| L(s) = 1 | + 25·5-s − 234·7-s + 347·11-s − 33·13-s + 237·17-s + 1.49e3·19-s − 2.81e3·23-s + 625·25-s − 5.51e3·29-s + 2.91e3·31-s − 5.85e3·35-s + 5.60e3·37-s + 4.71e3·41-s + 1.04e4·43-s + 5.96e3·47-s + 3.79e4·49-s − 1.79e4·53-s + 8.67e3·55-s + 3.03e4·59-s − 3.55e4·61-s − 825·65-s − 1.24e4·67-s + 7.52e3·71-s + 3.63e4·73-s − 8.11e4·77-s − 2.27e4·79-s + 4.62e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.80·7-s + 0.864·11-s − 0.0541·13-s + 0.198·17-s + 0.950·19-s − 1.10·23-s + 1/5·25-s − 1.21·29-s + 0.544·31-s − 0.807·35-s + 0.672·37-s + 0.438·41-s + 0.864·43-s + 0.393·47-s + 2.25·49-s − 0.878·53-s + 0.386·55-s + 1.13·59-s − 1.22·61-s − 0.0242·65-s − 0.339·67-s + 0.177·71-s + 0.798·73-s − 1.56·77-s − 0.409·79-s + 0.736·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 234 T + p^{5} T^{2} \) |
| 11 | \( 1 - 347 T + p^{5} T^{2} \) |
| 13 | \( 1 + 33 T + p^{5} T^{2} \) |
| 17 | \( 1 - 237 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1496 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2811 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5513 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2911 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5602 T + p^{5} T^{2} \) |
| 41 | \( 1 - 4716 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10479 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5963 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17964 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30372 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35530 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12476 T + p^{5} T^{2} \) |
| 71 | \( 1 - 7520 T + p^{5} T^{2} \) |
| 73 | \( 1 - 36378 T + p^{5} T^{2} \) |
| 79 | \( 1 + 22727 T + p^{5} T^{2} \) |
| 83 | \( 1 - 46254 T + p^{5} T^{2} \) |
| 89 | \( 1 + 58832 T + p^{5} T^{2} \) |
| 97 | \( 1 + 145906 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.097939515311542447195151765023, −7.81523232578969914992187772138, −6.91896988159313213859541350913, −6.17847617541267733123498598133, −5.60481081737031640810194253204, −4.15429481309921995968474263381, −3.38942587383030071762017549956, −2.44991101315731285001679188035, −1.12714601808032824039813874305, 0,
1.12714601808032824039813874305, 2.44991101315731285001679188035, 3.38942587383030071762017549956, 4.15429481309921995968474263381, 5.60481081737031640810194253204, 6.17847617541267733123498598133, 6.91896988159313213859541350913, 7.81523232578969914992187772138, 9.097939515311542447195151765023
