| L(s) = 1 | − 2.06e10·2-s + 2.80e20·4-s − 4.26e23·5-s − 3.03e28·7-s − 2.74e30·8-s + 8.83e33·10-s + 3.14e34·11-s + 2.32e37·13-s + 6.27e38·14-s + 1.54e40·16-s − 2.99e41·17-s − 2.19e42·19-s − 1.19e44·20-s − 6.49e44·22-s − 5.39e45·23-s + 1.14e47·25-s − 4.80e47·26-s − 8.50e48·28-s − 1.03e48·29-s − 1.03e49·31-s + 8.58e49·32-s + 6.19e51·34-s + 1.29e52·35-s − 7.45e51·37-s + 4.53e52·38-s + 1.17e54·40-s − 1.59e54·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.89·4-s − 1.64·5-s − 1.48·7-s − 1.53·8-s + 2.79·10-s + 0.407·11-s + 1.11·13-s + 2.52·14-s + 0.709·16-s − 1.80·17-s − 0.318·19-s − 3.11·20-s − 0.694·22-s − 1.30·23-s + 1.69·25-s − 1.90·26-s − 2.81·28-s − 0.105·29-s − 0.113·31-s + 0.324·32-s + 3.07·34-s + 2.43·35-s − 0.217·37-s + 0.542·38-s + 2.51·40-s − 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(68-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+67/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(34)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{69}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 2.06e10T + 1.47e20T^{2} \) |
| 5 | \( 1 + 4.26e23T + 6.77e46T^{2} \) |
| 7 | \( 1 + 3.03e28T + 4.18e56T^{2} \) |
| 11 | \( 1 - 3.14e34T + 5.93e69T^{2} \) |
| 13 | \( 1 - 2.32e37T + 4.30e74T^{2} \) |
| 17 | \( 1 + 2.99e41T + 2.75e82T^{2} \) |
| 19 | \( 1 + 2.19e42T + 4.74e85T^{2} \) |
| 23 | \( 1 + 5.39e45T + 1.72e91T^{2} \) |
| 29 | \( 1 + 1.03e48T + 9.56e97T^{2} \) |
| 31 | \( 1 + 1.03e49T + 8.34e99T^{2} \) |
| 37 | \( 1 + 7.45e51T + 1.17e105T^{2} \) |
| 41 | \( 1 + 1.59e54T + 1.13e108T^{2} \) |
| 43 | \( 1 + 3.44e54T + 2.76e109T^{2} \) |
| 47 | \( 1 + 1.39e56T + 1.07e112T^{2} \) |
| 53 | \( 1 - 7.64e57T + 3.36e115T^{2} \) |
| 59 | \( 1 + 3.63e58T + 4.43e118T^{2} \) |
| 61 | \( 1 - 4.97e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 2.19e61T + 2.22e122T^{2} \) |
| 71 | \( 1 + 1.04e62T + 1.08e124T^{2} \) |
| 73 | \( 1 - 2.78e62T + 6.96e124T^{2} \) |
| 79 | \( 1 + 3.71e63T + 1.38e127T^{2} \) |
| 83 | \( 1 - 3.08e62T + 3.78e128T^{2} \) |
| 89 | \( 1 - 1.40e65T + 4.06e130T^{2} \) |
| 97 | \( 1 + 1.11e66T + 1.29e133T^{2} \) |
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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
−9.991234372683511084634234875097, −8.806521180302004111712927753228, −8.259202419878825588500803743857, −6.94928258749478155496907279583, −6.43769448875175994557943182574, −4.12290365639855346115663138286, −3.30491312264480934736955914493, −1.93988397292900098097790953724, −0.56587818226941821077479736495, 0,
0.56587818226941821077479736495, 1.93988397292900098097790953724, 3.30491312264480934736955914493, 4.12290365639855346115663138286, 6.43769448875175994557943182574, 6.94928258749478155496907279583, 8.259202419878825588500803743857, 8.806521180302004111712927753228, 9.991234372683511084634234875097
