| L(s) = 1 | − 32.7·2-s + 214.·3-s + 560.·4-s − 7.03e3·6-s − 2.06e3·7-s − 1.58e3·8-s + 2.64e4·9-s + 4.21e4·11-s + 1.20e5·12-s − 2.85e4·13-s + 6.76e4·14-s − 2.35e5·16-s + 6.08e5·17-s − 8.66e5·18-s − 9.04e5·19-s − 4.44e5·21-s − 1.38e6·22-s − 3.24e5·23-s − 3.39e5·24-s + 9.35e5·26-s + 1.45e6·27-s − 1.15e6·28-s − 5.82e6·29-s − 2.41e6·31-s + 8.50e6·32-s + 9.05e6·33-s − 1.99e7·34-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.53·3-s + 1.09·4-s − 2.21·6-s − 0.325·7-s − 0.136·8-s + 1.34·9-s + 0.868·11-s + 1.67·12-s − 0.277·13-s + 0.470·14-s − 0.896·16-s + 1.76·17-s − 1.94·18-s − 1.59·19-s − 0.498·21-s − 1.25·22-s − 0.242·23-s − 0.208·24-s + 0.401·26-s + 0.527·27-s − 0.356·28-s − 1.52·29-s − 0.469·31-s + 1.43·32-s + 1.32·33-s − 2.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 32.7T + 512T^{2} \) |
| 3 | \( 1 - 214.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 2.06e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.21e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 6.08e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.24e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.82e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.41e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.56e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.80e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.16e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.64e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.28e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.99e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.17e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.21e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.57e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.73e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.21e8T + 7.60e17T^{2} \) |
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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
−9.327376337940609988802626786012, −8.858300364879757235269393013005, −7.85470194285290445745749469307, −7.40571963015357681174792504507, −6.12348405972583738089090446557, −4.24828744981561416141127404546, −3.26353712329493236099369073592, −2.11080314679830447431929471533, −1.33665246284576406711776498995, 0,
1.33665246284576406711776498995, 2.11080314679830447431929471533, 3.26353712329493236099369073592, 4.24828744981561416141127404546, 6.12348405972583738089090446557, 7.40571963015357681174792504507, 7.85470194285290445745749469307, 8.858300364879757235269393013005, 9.327376337940609988802626786012
