L(s) = 1 | − 23.9·2-s + 161.·3-s + 61.9·4-s + 2.64e3·5-s − 3.88e3·6-s + 2.05e3·7-s + 1.07e4·8-s + 6.55e3·9-s − 6.33e4·10-s − 3.15e3·11-s + 1.00e4·12-s − 1.10e5·13-s − 4.93e4·14-s + 4.28e5·15-s − 2.90e5·16-s + 5.71e5·17-s − 1.57e5·18-s − 1.94e5·19-s + 1.63e5·20-s + 3.33e5·21-s + 7.55e4·22-s + 2.29e6·23-s + 1.74e6·24-s + 5.03e6·25-s + 2.64e6·26-s − 2.12e6·27-s + 1.27e5·28-s + ⋯ |
L(s) = 1 | − 1.05·2-s + 1.15·3-s + 0.120·4-s + 1.89·5-s − 1.22·6-s + 0.324·7-s + 0.930·8-s + 0.333·9-s − 2.00·10-s − 0.0649·11-s + 0.139·12-s − 1.07·13-s − 0.343·14-s + 2.18·15-s − 1.10·16-s + 1.65·17-s − 0.352·18-s − 0.342·19-s + 0.228·20-s + 0.374·21-s + 0.0687·22-s + 1.71·23-s + 1.07·24-s + 2.57·25-s + 1.13·26-s − 0.770·27-s + 0.0392·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.040081574\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.040081574\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 - 7.07e5T \) |
good | 2 | \( 1 + 23.9T + 512T^{2} \) |
| 3 | \( 1 - 161.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.64e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.05e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.15e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.10e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.71e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.94e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.29e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 2.66e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.41e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.78e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.79e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.44e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.86e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.71e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.78e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.92e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.60e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.66e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.69e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.11e8T + 7.60e17T^{2} \) |
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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
−14.63642534258086066260330461321, −14.03911368628522366595306933861, −12.91811208040575172320701006024, −10.44983500976842003644628828185, −9.549409836124579618215026153797, −8.810453170826064980555535131289, −7.39067670630845850898336697816, −5.28608430048301561097855406634, −2.62136860136327137453231313146, −1.36710767826383268726192808894,
1.36710767826383268726192808894, 2.62136860136327137453231313146, 5.28608430048301561097855406634, 7.39067670630845850898336697816, 8.810453170826064980555535131289, 9.549409836124579618215026153797, 10.44983500976842003644628828185, 12.91811208040575172320701006024, 14.03911368628522366595306933861, 14.63642534258086066260330461321