| L(s) = 1 | − 2·3-s − 25·5-s − 62·7-s − 239·9-s − 144·11-s − 654·13-s + 50·15-s − 1.19e3·17-s + 556·19-s + 124·21-s + 2.18e3·23-s + 625·25-s + 964·27-s − 1.57e3·29-s + 9.66e3·31-s + 288·33-s + 1.55e3·35-s − 3.53e3·37-s + 1.30e3·39-s + 7.46e3·41-s − 7.11e3·43-s + 5.97e3·45-s − 2.82e4·47-s − 1.29e4·49-s + 2.38e3·51-s − 1.30e4·53-s + 3.60e3·55-s + ⋯ |
| L(s) = 1 | − 0.128·3-s − 0.447·5-s − 0.478·7-s − 0.983·9-s − 0.358·11-s − 1.07·13-s + 0.0573·15-s − 0.998·17-s + 0.353·19-s + 0.0613·21-s + 0.860·23-s + 1/5·25-s + 0.254·27-s − 0.348·29-s + 1.80·31-s + 0.0460·33-s + 0.213·35-s − 0.424·37-s + 0.137·39-s + 0.693·41-s − 0.586·43-s + 0.439·45-s − 1.86·47-s − 0.771·49-s + 0.128·51-s − 0.637·53-s + 0.160·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 2 T + p^{5} T^{2} \) |
| 7 | \( 1 + 62 T + p^{5} T^{2} \) |
| 11 | \( 1 + 144 T + p^{5} T^{2} \) |
| 13 | \( 1 + 654 T + p^{5} T^{2} \) |
| 17 | \( 1 + 70 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 556 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2182 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1578 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9660 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3534 T + p^{5} T^{2} \) |
| 41 | \( 1 - 182 p T + p^{5} T^{2} \) |
| 43 | \( 1 + 7114 T + p^{5} T^{2} \) |
| 47 | \( 1 + 602 p T + p^{5} T^{2} \) |
| 53 | \( 1 + 13046 T + p^{5} T^{2} \) |
| 59 | \( 1 + 37092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39570 T + p^{5} T^{2} \) |
| 67 | \( 1 + 56734 T + p^{5} T^{2} \) |
| 71 | \( 1 - 45588 T + p^{5} T^{2} \) |
| 73 | \( 1 - 11842 T + p^{5} T^{2} \) |
| 79 | \( 1 - 94216 T + p^{5} T^{2} \) |
| 83 | \( 1 + 31482 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94054 T + p^{5} T^{2} \) |
| 97 | \( 1 - 23714 T + p^{5} T^{2} \) |
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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
−14.65264907780091202240701549010, −13.33389242366691674483497162022, −12.08522562595288079709291406907, −11.01408617827620803938838983414, −9.536551314822357469633093740178, −8.140298728066282962127244961917, −6.62932334937497761230981415628, −4.92742681402920521918268040885, −2.89338442678784940824697414539, 0,
2.89338442678784940824697414539, 4.92742681402920521918268040885, 6.62932334937497761230981415628, 8.140298728066282962127244961917, 9.536551314822357469633093740178, 11.01408617827620803938838983414, 12.08522562595288079709291406907, 13.33389242366691674483497162022, 14.65264907780091202240701549010
