| L(s) = 1 | + 6·3-s − 4·5-s − 4·7-s + 27·9-s − 28·11-s − 26·13-s − 24·15-s − 36·17-s − 44·19-s − 24·21-s + 8·23-s − 226·25-s + 108·27-s − 204·29-s + 164·31-s − 168·33-s + 16·35-s − 668·37-s − 156·39-s − 100·41-s + 272·43-s − 108·45-s − 60·47-s − 566·49-s − 216·51-s − 708·53-s + 112·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.357·5-s − 0.215·7-s + 9-s − 0.767·11-s − 0.554·13-s − 0.413·15-s − 0.513·17-s − 0.531·19-s − 0.249·21-s + 0.0725·23-s − 1.80·25-s + 0.769·27-s − 1.30·29-s + 0.950·31-s − 0.886·33-s + 0.0772·35-s − 2.96·37-s − 0.640·39-s − 0.380·41-s + 0.964·43-s − 0.357·45-s − 0.186·47-s − 1.65·49-s − 0.593·51-s − 1.83·53-s + 0.274·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 242 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 582 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 28 T + 2426 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36 T + 2038 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 44 T - 498 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 204 T + 52270 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 164 T + 66294 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 668 T + 212814 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T - 2230 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 272 T + 137142 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 60 T + 203746 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 708 T + 312478 T^{2} + 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 180 T + 395626 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1068 T + 726830 T^{2} + 1068 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 420 T + 207854 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 436 T + 749474 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 412 T + 163398 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 672 T + 559646 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 124 T + 501530 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 140 T + 1088138 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 188 T - 343530 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.789258739575076852169930477637, −9.694363709486846408786170940517, −8.954145579524072772818608161960, −8.792655390419103954979094097120, −8.148080423537310633559375186704, −7.88238360860659626711853741046, −7.35033661856113913683140544656, −7.19256751390995337792015543616, −6.23462255446169174667819699184, −6.19793003563150396067527806091, −5.12286682976149393526029596432, −4.96405277157720327117499920335, −4.18058544513800717697195864857, −3.77054670889504364859528607188, −3.17571619151732251100228736435, −2.72714156116705116800813935824, −1.88577486887734932884279074016, −1.66822073065225087140835287046, 0, 0,
1.66822073065225087140835287046, 1.88577486887734932884279074016, 2.72714156116705116800813935824, 3.17571619151732251100228736435, 3.77054670889504364859528607188, 4.18058544513800717697195864857, 4.96405277157720327117499920335, 5.12286682976149393526029596432, 6.19793003563150396067527806091, 6.23462255446169174667819699184, 7.19256751390995337792015543616, 7.35033661856113913683140544656, 7.88238360860659626711853741046, 8.148080423537310633559375186704, 8.792655390419103954979094097120, 8.954145579524072772818608161960, 9.694363709486846408786170940517, 9.789258739575076852169930477637
