L(s) = 1 | + 6.73·2-s − 9·3-s + 13.3·4-s − 60.5·6-s − 2.41·7-s − 125.·8-s + 81·9-s + 121·11-s − 119.·12-s − 334.·13-s − 16.2·14-s − 1.27e3·16-s + 1.21e3·17-s + 545.·18-s − 746.·19-s + 21.7·21-s + 814.·22-s − 3.38e3·23-s + 1.13e3·24-s − 2.25e3·26-s − 729·27-s − 32.0·28-s − 2.01e3·29-s + 8.68e3·31-s − 4.53e3·32-s − 1.08e3·33-s + 8.15e3·34-s + ⋯ |
L(s) = 1 | + 1.18·2-s − 0.577·3-s + 0.415·4-s − 0.686·6-s − 0.0186·7-s − 0.695·8-s + 0.333·9-s + 0.301·11-s − 0.240·12-s − 0.549·13-s − 0.0221·14-s − 1.24·16-s + 1.01·17-s + 0.396·18-s − 0.474·19-s + 0.0107·21-s + 0.358·22-s − 1.33·23-s + 0.401·24-s − 0.654·26-s − 0.192·27-s − 0.00773·28-s − 0.444·29-s + 1.62·31-s − 0.783·32-s − 0.174·33-s + 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.485494687\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.485494687\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 - 6.73T + 32T^{2} \) |
| 7 | \( 1 + 2.41T + 1.68e4T^{2} \) |
| 13 | \( 1 + 334.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 746.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.83e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.81e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 986.T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.41e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.38e5T + 8.58e9T^{2} \) |
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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
−9.705690957217409601660428683397, −8.587187257558176388389102405967, −7.56219068820089971366503248351, −6.47112693626538797219482192584, −5.86859049591688145793167204393, −4.98822739285935625765675501262, −4.22003627880213352014546946565, −3.31531646537571454467197333258, −2.10754842471413341230367386340, −0.60045209613967749526446907242,
0.60045209613967749526446907242, 2.10754842471413341230367386340, 3.31531646537571454467197333258, 4.22003627880213352014546946565, 4.98822739285935625765675501262, 5.86859049591688145793167204393, 6.47112693626538797219482192584, 7.56219068820089971366503248351, 8.587187257558176388389102405967, 9.705690957217409601660428683397