| L(s) = 1 | + 2·2-s − 4·3-s + 2·4-s − 8·6-s + 2·7-s + 5·8-s + 13·9-s + 3·11-s − 8·12-s − 9·13-s + 4·14-s + 5·16-s + 7·17-s + 26·18-s + 5·19-s − 8·21-s + 6·22-s − 9·23-s − 20·24-s − 18·26-s − 30·27-s + 4·28-s − 2·31-s − 2·32-s − 12·33-s + 14·34-s + 26·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 4-s − 3.26·6-s + 0.755·7-s + 1.76·8-s + 13/3·9-s + 0.904·11-s − 2.30·12-s − 2.49·13-s + 1.06·14-s + 5/4·16-s + 1.69·17-s + 6.12·18-s + 1.14·19-s − 1.74·21-s + 1.27·22-s − 1.87·23-s − 4.08·24-s − 3.53·26-s − 5.77·27-s + 0.755·28-s − 0.359·31-s − 0.353·32-s − 2.08·33-s + 2.40·34-s + 13/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.909590826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.909590826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2:C_4$ | \( 1 + 4 T + p T^{2} - 10 T^{3} - 29 T^{4} - 10 p T^{5} + p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2:C_4$ | \( 1 + 9 T + 48 T^{2} + 235 T^{3} + 1011 T^{4} + 235 p T^{5} + 48 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 7 T + 52 T^{2} - 245 T^{3} + 1311 T^{4} - 245 p T^{5} + 52 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 9 T + 13 T^{2} - 15 T^{3} + 196 T^{4} - 15 p T^{5} + 13 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 11 T^{2} + 90 T^{3} + 661 T^{4} + 90 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 116 p T^{5} - 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 356 p T^{5} + 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 13 T + 67 T^{2} - 115 T^{3} - 3864 T^{4} - 115 p T^{5} + 67 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 14 T + 23 T^{2} - 400 T^{3} - 2899 T^{4} - 400 p T^{5} + 23 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 15 T + 206 T^{2} + 1965 T^{3} + 18601 T^{4} + 1965 p T^{5} + 206 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 17 T + 42 T^{2} + 1025 T^{3} - 12559 T^{4} + 1025 p T^{5} + 42 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 13 T + 133 T^{2} - 1541 T^{3} + 17940 T^{4} - 1541 p T^{5} + 133 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 11 T - 27 T^{2} + 515 T^{3} - 364 T^{4} + 515 p T^{5} - 27 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 15 T + 56 T^{2} - 675 T^{3} + 11821 T^{4} - 675 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 24 T + 173 T^{2} + 360 T^{3} + 361 T^{4} + 360 p T^{5} + 173 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 79 T^{2} - 420 T^{3} + 7501 T^{4} - 420 p T^{5} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 12 T - 3 T^{2} + 1220 T^{3} - 13419 T^{4} + 1220 p T^{5} - 3 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47836318088706972732893084734, −7.33600057268098509361781271926, −7.27065474882939287827323978196, −6.82931178072249117282980337294, −6.56294181489535797219594877233, −6.30514150552568834721157066857, −6.26635016774119225323197472347, −5.91599592953959347989224756091, −5.52405395929281706886118680218, −5.29068520051448658715331020709, −5.25351622840027320775193644875, −4.85104492359748857139906578976, −4.75503919582521384755757918915, −4.64714061143781013976389274238, −4.32841469839764762129920312591, −4.04707876825926941313672645168, −3.88777033794191258070430882843, −3.35623568282945456974879732985, −3.25482761404315604856614474424, −2.61463517132116180693321874054, −1.88265087388517799331158264316, −1.86671081774407816360712955642, −1.80619808202100508086949096656, −0.894556500888698653190925372447, −0.67382017456348229249607518921,
0.67382017456348229249607518921, 0.894556500888698653190925372447, 1.80619808202100508086949096656, 1.86671081774407816360712955642, 1.88265087388517799331158264316, 2.61463517132116180693321874054, 3.25482761404315604856614474424, 3.35623568282945456974879732985, 3.88777033794191258070430882843, 4.04707876825926941313672645168, 4.32841469839764762129920312591, 4.64714061143781013976389274238, 4.75503919582521384755757918915, 4.85104492359748857139906578976, 5.25351622840027320775193644875, 5.29068520051448658715331020709, 5.52405395929281706886118680218, 5.91599592953959347989224756091, 6.26635016774119225323197472347, 6.30514150552568834721157066857, 6.56294181489535797219594877233, 6.82931178072249117282980337294, 7.27065474882939287827323978196, 7.33600057268098509361781271926, 7.47836318088706972732893084734
