| L(s) = 1 | + 16·2-s + 256·4-s − 435·5-s − 2.52e3·7-s + 4.09e3·8-s − 6.96e3·10-s − 9.12e3·11-s − 7.91e4·13-s − 4.04e4·14-s + 6.55e4·16-s − 4.37e5·17-s + 1.16e5·19-s − 1.11e5·20-s − 1.45e5·22-s − 2.61e5·23-s − 1.76e6·25-s − 1.26e6·26-s − 6.46e5·28-s − 3.96e5·29-s − 5.88e6·31-s + 1.04e6·32-s − 7.00e6·34-s + 1.09e6·35-s − 8.98e6·37-s + 1.87e6·38-s − 1.78e6·40-s + 1.74e7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.311·5-s − 0.397·7-s + 0.353·8-s − 0.220·10-s − 0.187·11-s − 0.768·13-s − 0.281·14-s + 1/4·16-s − 1.27·17-s + 0.205·19-s − 0.155·20-s − 0.132·22-s − 0.194·23-s − 0.903·25-s − 0.543·26-s − 0.198·28-s − 0.104·29-s − 1.14·31-s + 0.176·32-s − 0.899·34-s + 0.123·35-s − 0.788·37-s + 0.145·38-s − 0.110·40-s + 0.964·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 87 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 361 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 9123 T + p^{9} T^{2} \) |
| 13 | \( 1 + 79180 T + p^{9} T^{2} \) |
| 17 | \( 1 + 437976 T + p^{9} T^{2} \) |
| 19 | \( 1 - 116966 T + p^{9} T^{2} \) |
| 23 | \( 1 + 261102 T + p^{9} T^{2} \) |
| 29 | \( 1 + 396150 T + p^{9} T^{2} \) |
| 31 | \( 1 + 5882533 T + p^{9} T^{2} \) |
| 37 | \( 1 + 8986246 T + p^{9} T^{2} \) |
| 41 | \( 1 - 17449566 T + p^{9} T^{2} \) |
| 43 | \( 1 + 32094646 T + p^{9} T^{2} \) |
| 47 | \( 1 - 20965782 T + p^{9} T^{2} \) |
| 53 | \( 1 - 40669047 T + p^{9} T^{2} \) |
| 59 | \( 1 + 84383076 T + p^{9} T^{2} \) |
| 61 | \( 1 + 148038424 T + p^{9} T^{2} \) |
| 67 | \( 1 - 154939106 T + p^{9} T^{2} \) |
| 71 | \( 1 - 168343560 T + p^{9} T^{2} \) |
| 73 | \( 1 - 418697993 T + p^{9} T^{2} \) |
| 79 | \( 1 - 210598040 T + p^{9} T^{2} \) |
| 83 | \( 1 - 776394525 T + p^{9} T^{2} \) |
| 89 | \( 1 - 370837746 T + p^{9} T^{2} \) |
| 97 | \( 1 - 309841967 T + p^{9} T^{2} \) |
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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
−12.88372017195797234519055815524, −11.86610457897130064965403803914, −10.72047428595499673896912644197, −9.331204129440178173824856587120, −7.71538397990724234971668049271, −6.50706422120862006957101673322, −5.04505389804521224189835032042, −3.69170885157970475903885028831, −2.17699802954477777202364576528, 0,
2.17699802954477777202364576528, 3.69170885157970475903885028831, 5.04505389804521224189835032042, 6.50706422120862006957101673322, 7.71538397990724234971668049271, 9.331204129440178173824856587120, 10.72047428595499673896912644197, 11.86610457897130064965403803914, 12.88372017195797234519055815524
