| L(s) = 1 | − 2·2-s + 3·3-s + 2·5-s − 6·6-s + 8·8-s − 4·10-s + 8·11-s + 84·13-s + 6·15-s − 16·16-s − 2·17-s − 124·19-s − 16·22-s − 76·23-s + 24·24-s + 125·25-s − 168·26-s − 27·27-s + 508·29-s − 12·30-s − 72·31-s + 24·33-s + 4·34-s − 398·37-s + 248·38-s + 252·39-s + 16·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.178·5-s − 0.408·6-s + 0.353·8-s − 0.126·10-s + 0.219·11-s + 1.79·13-s + 0.103·15-s − 1/4·16-s − 0.0285·17-s − 1.49·19-s − 0.155·22-s − 0.689·23-s + 0.204·24-s + 25-s − 1.26·26-s − 0.192·27-s + 3.25·29-s − 0.0730·30-s − 0.417·31-s + 0.126·33-s + 0.0201·34-s − 1.76·37-s + 1.05·38-s + 1.03·39-s + 0.0632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.951812414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.951812414\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - 121 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T - 1267 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 4909 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 124 T + 8517 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 76 T - 6391 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 72 T - 24607 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 462 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 264 T - 34127 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 162 T - 122633 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 772 T + 390605 T^{2} + 772 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 30 T - 226081 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 764 T + 282933 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 236 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 418 T - 214293 T^{2} - 418 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 552 T - 188335 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1036 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 30 T - 704069 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )^{2} \) |
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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
−11.53725479286600121362367391833, −10.90115468142789234651845206254, −10.44471923463469963344839612632, −10.23954332480614220863601933821, −9.781817768586180554706674978571, −8.801693925834719478637658165565, −8.663283347860548424563703355135, −8.523027975413870497176806414116, −8.125301732646865638181018488672, −7.11413207322669357431282959976, −6.79230927623040578910968057383, −6.23960926425727334417518347576, −5.79950120912725062992672423070, −4.64152918474449824111724886675, −4.61162398648325888743969853481, −3.42801256407147671091014261862, −3.26817344681815310045395079054, −2.09698476953691718855373320840, −1.51807179469123552343977904843, −0.59434454809991883647781897513,
0.59434454809991883647781897513, 1.51807179469123552343977904843, 2.09698476953691718855373320840, 3.26817344681815310045395079054, 3.42801256407147671091014261862, 4.61162398648325888743969853481, 4.64152918474449824111724886675, 5.79950120912725062992672423070, 6.23960926425727334417518347576, 6.79230927623040578910968057383, 7.11413207322669357431282959976, 8.125301732646865638181018488672, 8.523027975413870497176806414116, 8.663283347860548424563703355135, 8.801693925834719478637658165565, 9.781817768586180554706674978571, 10.23954332480614220863601933821, 10.44471923463469963344839612632, 10.90115468142789234651845206254, 11.53725479286600121362367391833
