| L(s) = 1 | + 2.18·2-s + 2.77·4-s + 3.43·5-s + 3.93·7-s + 1.69·8-s + 7.49·10-s − 11-s + 3.31·13-s + 8.60·14-s − 1.84·16-s − 2.80·17-s − 19-s + 9.52·20-s − 2.18·22-s − 6.88·23-s + 6.77·25-s + 7.23·26-s + 10.9·28-s − 5.67·29-s + 2.51·31-s − 7.42·32-s − 6.13·34-s + 13.5·35-s − 6.39·37-s − 2.18·38-s + 5.81·40-s − 0.560·41-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.38·4-s + 1.53·5-s + 1.48·7-s + 0.598·8-s + 2.37·10-s − 0.301·11-s + 0.918·13-s + 2.30·14-s − 0.462·16-s − 0.680·17-s − 0.229·19-s + 2.12·20-s − 0.465·22-s − 1.43·23-s + 1.35·25-s + 1.41·26-s + 2.06·28-s − 1.05·29-s + 0.452·31-s − 1.31·32-s − 1.05·34-s + 2.28·35-s − 1.05·37-s − 0.354·38-s + 0.919·40-s − 0.0875·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.901925976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.901925976\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.18T + 2T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 23 | \( 1 + 6.88T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 + 0.560T + 41T^{2} \) |
| 43 | \( 1 + 9.40T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 5.68T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + 9.95T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 - 8.49T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−9.147682039713645139706708897810, −8.474167875518717747717093912203, −7.44315232297323850331442867219, −6.31042616935053828486743537464, −5.90754215271266643690424027762, −5.12696848381543459067539136106, −4.52085708578687058454735564483, −3.50562973823269645177866913958, −2.17569141872391864010893096727, −1.76359882748356636509338428509,
1.76359882748356636509338428509, 2.17569141872391864010893096727, 3.50562973823269645177866913958, 4.52085708578687058454735564483, 5.12696848381543459067539136106, 5.90754215271266643690424027762, 6.31042616935053828486743537464, 7.44315232297323850331442867219, 8.474167875518717747717093912203, 9.147682039713645139706708897810
