| L(s) = 1 | − 21.2·2-s − 64.4·3-s − 61.1·4-s − 398.·5-s + 1.36e3·6-s + 3.85e3·7-s + 1.21e4·8-s − 1.55e4·9-s + 8.45e3·10-s + 5.78e4·11-s + 3.94e3·12-s + 1.88e4·13-s − 8.17e4·14-s + 2.56e4·15-s − 2.27e5·16-s + 3.29e5·18-s − 7.86e5·19-s + 2.43e4·20-s − 2.48e5·21-s − 1.22e6·22-s + 8.21e5·23-s − 7.84e5·24-s − 1.79e6·25-s − 4.00e5·26-s + 2.26e6·27-s − 2.35e5·28-s + 1.71e6·29-s + ⋯ |
| L(s) = 1 | − 0.938·2-s − 0.459·3-s − 0.119·4-s − 0.285·5-s + 0.431·6-s + 0.606·7-s + 1.05·8-s − 0.788·9-s + 0.267·10-s + 1.19·11-s + 0.0549·12-s + 0.183·13-s − 0.568·14-s + 0.130·15-s − 0.866·16-s + 0.740·18-s − 1.38·19-s + 0.0340·20-s − 0.278·21-s − 1.11·22-s + 0.612·23-s − 0.482·24-s − 0.918·25-s − 0.172·26-s + 0.821·27-s − 0.0724·28-s + 0.449·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 21.2T + 512T^{2} \) |
| 3 | \( 1 + 64.4T + 1.96e4T^{2} \) |
| 5 | \( 1 + 398.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.85e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.78e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.88e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 7.86e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.21e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.71e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.63e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.29e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.40e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.23e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.62e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.02e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.74e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.48e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.56e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.37e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.12e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.46e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.36e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.21e9T + 7.60e17T^{2} \) |
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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
−9.585442386555641491534990566308, −8.760012503429528095275075518857, −8.145772057572097426044817603638, −7.01096672078252106539139963264, −5.92112966463488094747369932117, −4.71694453063106571200122816685, −3.79330921812401348035541507137, −2.04117005284598091547932754689, −0.960844777894605493950212261821, 0,
0.960844777894605493950212261821, 2.04117005284598091547932754689, 3.79330921812401348035541507137, 4.71694453063106571200122816685, 5.92112966463488094747369932117, 7.01096672078252106539139963264, 8.145772057572097426044817603638, 8.760012503429528095275075518857, 9.585442386555641491534990566308
