| L(s) = 1 | + 300·5-s − 243·9-s + 6.78e3·13-s + 1.03e4·17-s + 3.62e4·25-s + 6.42e4·29-s − 1.52e5·37-s − 1.40e5·41-s − 7.29e4·45-s + 1.29e5·49-s + 1.33e5·53-s − 5.14e5·61-s + 2.03e6·65-s + 4.86e5·73-s + 5.90e4·81-s + 3.10e6·85-s − 1.37e6·89-s − 1.88e6·97-s − 5.19e5·101-s − 2.04e6·109-s − 2.62e6·113-s − 1.64e6·117-s + 1.36e6·121-s − 5.62e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 12/5·5-s − 1/3·9-s + 3.08·13-s + 2.10·17-s + 2.31·25-s + 2.63·29-s − 3.00·37-s − 2.03·41-s − 4/5·45-s + 1.09·49-s + 0.899·53-s − 2.26·61-s + 7.41·65-s + 1.25·73-s + 1/9·81-s + 5.05·85-s − 1.94·89-s − 2.06·97-s − 0.504·101-s − 1.58·109-s − 1.82·113-s − 1.02·117-s + 0.770·121-s − 0.287·125-s + 6.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.914090654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.914090654\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 6 p^{2} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 129266 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1365410 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3394 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5178 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 47709890 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 280146530 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 32142 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 711526610 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 76150 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 70038 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2451652130 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1425021694 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 66942 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 68178858910 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 257014 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 77748332930 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138474492194 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 243442 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 260585085842 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 415389237694 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 686766 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 942686 T + p^{6} T^{2} )^{2} \) |
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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
−14.25903235033281588114271412315, −13.89140620004396979767327969704, −13.64964898334371702673047700766, −13.43299135497336254025284132279, −12.23555806775937684344234477076, −12.05056755219228819680087407418, −10.82473273115840995688584584309, −10.41384330456026125450276741105, −10.07557653603962168161161801631, −9.284074084583963413651739528205, −8.531351005914113195887810404574, −8.303033359169925293591242609885, −6.75642421751498032990890837195, −6.31229452120245479164839927420, −5.59815303067403071130218220833, −5.34542572234888493888685663098, −3.69904105323943344519986996636, −2.92924760853123386070338710136, −1.51022245345459716902164341384, −1.28406475732437155487881840201,
1.28406475732437155487881840201, 1.51022245345459716902164341384, 2.92924760853123386070338710136, 3.69904105323943344519986996636, 5.34542572234888493888685663098, 5.59815303067403071130218220833, 6.31229452120245479164839927420, 6.75642421751498032990890837195, 8.303033359169925293591242609885, 8.531351005914113195887810404574, 9.284074084583963413651739528205, 10.07557653603962168161161801631, 10.41384330456026125450276741105, 10.82473273115840995688584584309, 12.05056755219228819680087407418, 12.23555806775937684344234477076, 13.43299135497336254025284132279, 13.64964898334371702673047700766, 13.89140620004396979767327969704, 14.25903235033281588114271412315
