| L(s) = 1 | + 5.00·2-s + 17.0·4-s − 22.0·5-s + 18.0·7-s + 45.3·8-s − 110.·10-s − 9.97·11-s − 19.0·13-s + 90.4·14-s + 90.3·16-s + 89.0·17-s − 50.9·19-s − 375.·20-s − 49.9·22-s − 3.14·23-s + 359.·25-s − 95.2·26-s + 308.·28-s − 157.·29-s − 203.·31-s + 89.7·32-s + 445.·34-s − 397.·35-s + 114.·37-s − 254.·38-s − 997.·40-s − 347.·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.13·4-s − 1.96·5-s + 0.975·7-s + 2.00·8-s − 3.48·10-s − 0.273·11-s − 0.406·13-s + 1.72·14-s + 1.41·16-s + 1.26·17-s − 0.614·19-s − 4.19·20-s − 0.483·22-s − 0.0284·23-s + 2.87·25-s − 0.718·26-s + 2.07·28-s − 1.01·29-s − 1.17·31-s + 0.495·32-s + 2.24·34-s − 1.92·35-s + 0.506·37-s − 1.08·38-s − 3.94·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 5.00T + 8T^{2} \) |
| 5 | \( 1 + 22.0T + 125T^{2} \) |
| 7 | \( 1 - 18.0T + 343T^{2} \) |
| 11 | \( 1 + 9.97T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 3.14T + 1.21e4T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 557.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 217.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 104.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 73.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 923.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 592.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 371.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714.T + 9.12e5T^{2} \) |
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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
−7.86785546104947342198978045714, −7.54375251002366180393862494266, −6.77806327856085784835399619381, −5.56994933790907395029955766981, −4.95131142954786364396413024379, −4.24320458846087502915039461237, −3.63322793538337260578968013781, −2.87383956091007343637355417802, −1.57454115564792177278158180182, 0,
1.57454115564792177278158180182, 2.87383956091007343637355417802, 3.63322793538337260578968013781, 4.24320458846087502915039461237, 4.95131142954786364396413024379, 5.56994933790907395029955766981, 6.77806327856085784835399619381, 7.54375251002366180393862494266, 7.86785546104947342198978045714
