| L(s) = 1 | − 0.372·3-s + 16.8·5-s − 3.51·7-s − 26.8·9-s + 21.1·11-s − 14.0·13-s − 6.27·15-s − 17.2·17-s + 19·19-s + 1.30·21-s − 171.·23-s + 159.·25-s + 20.0·27-s − 264.·29-s + 185.·31-s − 7.86·33-s − 59.1·35-s − 212.·37-s + 5.24·39-s − 157.·41-s + 258.·43-s − 452.·45-s − 293.·47-s − 330.·49-s + 6.41·51-s − 215.·53-s + 356.·55-s + ⋯ |
| L(s) = 1 | − 0.0716·3-s + 1.50·5-s − 0.189·7-s − 0.994·9-s + 0.579·11-s − 0.300·13-s − 0.108·15-s − 0.245·17-s + 0.229·19-s + 0.0135·21-s − 1.55·23-s + 1.27·25-s + 0.142·27-s − 1.69·29-s + 1.07·31-s − 0.0415·33-s − 0.285·35-s − 0.946·37-s + 0.0215·39-s − 0.598·41-s + 0.916·43-s − 1.50·45-s − 0.911·47-s − 0.964·49-s + 0.0176·51-s − 0.559·53-s + 0.873·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 0.372T + 27T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 7 | \( 1 + 3.51T + 343T^{2} \) |
| 11 | \( 1 - 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 215.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 147.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 913.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 678.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 608.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 282.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 214.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.67e3T + 9.12e5T^{2} \) |
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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
−9.068167005174113828773854213085, −8.319150169414500960058484808155, −7.16503793920054272441411870222, −6.13257288915913259097730484378, −5.83162928632214186650586225155, −4.83115157832770512854110309056, −3.53287340085859216951421719995, −2.42595116677216165574092280471, −1.59811692379856497312730613289, 0,
1.59811692379856497312730613289, 2.42595116677216165574092280471, 3.53287340085859216951421719995, 4.83115157832770512854110309056, 5.83162928632214186650586225155, 6.13257288915913259097730484378, 7.16503793920054272441411870222, 8.319150169414500960058484808155, 9.068167005174113828773854213085
