L(s) = 1 | − 3·2-s + 3·3-s + 4-s + 9·5-s − 9·6-s − 2·7-s + 21·8-s + 9·9-s − 27·10-s − 30·11-s + 3·12-s + 6·14-s + 27·15-s − 71·16-s − 111·17-s − 27·18-s + 46·19-s + 9·20-s − 6·21-s + 90·22-s − 6·23-s + 63·24-s − 44·25-s + 27·27-s − 2·28-s − 105·29-s − 81·30-s + ⋯ |
L(s) = 1 | − 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.804·5-s − 0.612·6-s − 0.107·7-s + 0.928·8-s + 1/3·9-s − 0.853·10-s − 0.822·11-s + 0.0721·12-s + 0.114·14-s + 0.464·15-s − 1.10·16-s − 1.58·17-s − 0.353·18-s + 0.555·19-s + 0.100·20-s − 0.0623·21-s + 0.872·22-s − 0.0543·23-s + 0.535·24-s − 0.351·25-s + 0.192·27-s − 0.0134·28-s − 0.672·29-s − 0.492·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - p T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 111 T + p^{3} T^{2} \) |
| 19 | \( 1 - 46 T + p^{3} T^{2} \) |
| 23 | \( 1 + 6 T + p^{3} T^{2} \) |
| 29 | \( 1 + 105 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 17 T + p^{3} T^{2} \) |
| 41 | \( 1 - 231 T + p^{3} T^{2} \) |
| 43 | \( 1 + 514 T + p^{3} T^{2} \) |
| 47 | \( 1 - 162 T + p^{3} T^{2} \) |
| 53 | \( 1 - 639 T + p^{3} T^{2} \) |
| 59 | \( 1 + 600 T + p^{3} T^{2} \) |
| 61 | \( 1 - 233 T + p^{3} T^{2} \) |
| 67 | \( 1 + 926 T + p^{3} T^{2} \) |
| 71 | \( 1 - 930 T + p^{3} T^{2} \) |
| 73 | \( 1 - 253 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1324 T + p^{3} T^{2} \) |
| 83 | \( 1 + 810 T + p^{3} T^{2} \) |
| 89 | \( 1 + 498 T + p^{3} T^{2} \) |
| 97 | \( 1 + 14 p T + p^{3} T^{2} \) |
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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
−9.866217548633004449106839787889, −9.194830966383675184165454641570, −8.451763235758352796712708115192, −7.60890377638021368282601318689, −6.65058606664874317616957796889, −5.35056723379727738465505340985, −4.23016663879819715368972859203, −2.62537180424150233080846076205, −1.61839528529427016298760100429, 0,
1.61839528529427016298760100429, 2.62537180424150233080846076205, 4.23016663879819715368972859203, 5.35056723379727738465505340985, 6.65058606664874317616957796889, 7.60890377638021368282601318689, 8.451763235758352796712708115192, 9.194830966383675184165454641570, 9.866217548633004449106839787889