| L(s) = 1 | + 3·2-s + 4·3-s + 7·4-s + 12·6-s − 12·7-s + 15·8-s + 13·9-s + 3·11-s + 28·12-s + 9·13-s − 36·14-s + 30·16-s + 3·17-s + 39·18-s − 48·21-s + 9·22-s + 4·23-s + 60·24-s + 27·26-s + 30·27-s − 84·28-s + 15·29-s + 13·31-s + 57·32-s + 12·33-s + 9·34-s + 91·36-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2.30·3-s + 7/2·4-s + 4.89·6-s − 4.53·7-s + 5.30·8-s + 13/3·9-s + 0.904·11-s + 8.08·12-s + 2.49·13-s − 9.62·14-s + 15/2·16-s + 0.727·17-s + 9.19·18-s − 10.4·21-s + 1.91·22-s + 0.834·23-s + 12.2·24-s + 5.29·26-s + 5.77·27-s − 15.8·28-s + 2.78·29-s + 2.33·31-s + 10.0·32-s + 2.08·33-s + 1.54·34-s + 91/6·36-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
| \(L(1)\) |
\(\approx\) |
\(33.93643765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(33.93643765\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 - 3 T - 2 T^{2} + 39 T^{3} - 95 T^{4} + 39 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 15 p T^{5} + 23 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 3 T + 2 T^{2} - 75 T^{3} + 511 T^{4} - 75 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 9 T^{2} + 70 T^{3} + 291 T^{4} + 70 p T^{5} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 4 T - 7 T^{2} - 70 T^{3} + 821 T^{4} - 70 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 - 15 T + 106 T^{2} - 675 T^{3} + 4171 T^{4} - 675 p T^{5} + 106 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 13 T + 63 T^{2} - 11 p T^{3} + 80 p T^{4} - 11 p^{2} T^{5} + 63 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 12 T + 107 T^{2} + 870 T^{3} + 6661 T^{4} + 870 p T^{5} + 107 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 3 T - 32 T^{2} + 219 T^{3} + 655 T^{4} + 219 p T^{5} - 32 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 17 T + 137 T^{2} + 1145 T^{3} + 9596 T^{4} + 1145 p T^{5} + 137 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + T - 37 T^{2} - 305 T^{3} + 1976 T^{4} - 305 p T^{5} - 37 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 31 T^{2} - 210 T^{3} + 2851 T^{4} - 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - 29 T + 331 T^{2} - 29 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^2:C_4$ | \( 1 - 3 T + 77 T^{2} - 375 T^{3} + 3436 T^{4} - 375 p T^{5} + 77 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 6 T - 37 T^{2} + 300 T^{3} + 7381 T^{4} + 300 p T^{5} - 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 15 T + 6 T^{2} - 815 T^{3} - 5979 T^{4} - 815 p T^{5} + 6 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 14 T - 7 T^{2} + 640 T^{3} - 2059 T^{4} + 640 p T^{5} - 7 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 30 T + 451 T^{2} - 5400 T^{3} + 56341 T^{4} - 5400 p T^{5} + 451 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 765 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50684870469363063174175620980, −7.22381330968840907906094937720, −6.74456190970950968403419079276, −6.71130919378217510094103776679, −6.70686955884954272532537570585, −6.48111086092776349042563356192, −6.41145826605197682644871307941, −6.21748982557253763632163374867, −6.06711439472992073531994287364, −5.20585015792649268876063448044, −5.20137392361901889397841789226, −4.76201622512382784594372815926, −4.73386085956054442173962955082, −3.97619891696898043366659814084, −3.91238675239480184149859892589, −3.79123734299926871180519417573, −3.32049576332143229011766960735, −3.30873241719855616087609529590, −3.17431869568649634579266873876, −3.04827072211827463998116430552, −2.76177653555214641126305964775, −2.17891869295448590502724055618, −1.69067021465241943755698994487, −1.28017460048614906423799288955, −1.12748878967438659210837673130,
1.12748878967438659210837673130, 1.28017460048614906423799288955, 1.69067021465241943755698994487, 2.17891869295448590502724055618, 2.76177653555214641126305964775, 3.04827072211827463998116430552, 3.17431869568649634579266873876, 3.30873241719855616087609529590, 3.32049576332143229011766960735, 3.79123734299926871180519417573, 3.91238675239480184149859892589, 3.97619891696898043366659814084, 4.73386085956054442173962955082, 4.76201622512382784594372815926, 5.20137392361901889397841789226, 5.20585015792649268876063448044, 6.06711439472992073531994287364, 6.21748982557253763632163374867, 6.41145826605197682644871307941, 6.48111086092776349042563356192, 6.70686955884954272532537570585, 6.71130919378217510094103776679, 6.74456190970950968403419079276, 7.22381330968840907906094937720, 7.50684870469363063174175620980
