| L(s) = 1 | + 2·2-s + 0.296·3-s + 4·4-s − 5·5-s + 0.592·6-s − 21.3·7-s + 8·8-s − 26.9·9-s − 10·10-s − 65.5·11-s + 1.18·12-s − 0.271·13-s − 42.6·14-s − 1.48·15-s + 16·16-s + 132.·17-s − 53.8·18-s − 19·19-s − 20·20-s − 6.31·21-s − 131.·22-s − 142.·23-s + 2.36·24-s + 25·25-s − 0.542·26-s − 15.9·27-s − 85.2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0569·3-s + 0.5·4-s − 0.447·5-s + 0.0403·6-s − 1.15·7-s + 0.353·8-s − 0.996·9-s − 0.316·10-s − 1.79·11-s + 0.0284·12-s − 0.00578·13-s − 0.814·14-s − 0.0254·15-s + 0.250·16-s + 1.89·17-s − 0.704·18-s − 0.229·19-s − 0.223·20-s − 0.0656·21-s − 1.27·22-s − 1.29·23-s + 0.0201·24-s + 0.200·25-s − 0.00408·26-s − 0.113·27-s − 0.575·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 0.296T + 27T^{2} \) |
| 7 | \( 1 + 21.3T + 343T^{2} \) |
| 11 | \( 1 + 65.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.271T + 2.19e3T^{2} \) |
| 17 | \( 1 - 132.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 83.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 433.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 63.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 609.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 970.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 580.T + 9.12e5T^{2} \) |
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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
−11.92941000245895948639191161844, −10.62509361943127686838005164105, −9.893982389106184268672414672747, −8.313019769084536360193690639892, −7.49070531765642687188640723413, −6.04717505110430014358531778979, −5.26896084599482377731742830053, −3.56849576034648849055173411339, −2.70514652014912167584210517812, 0,
2.70514652014912167584210517812, 3.56849576034648849055173411339, 5.26896084599482377731742830053, 6.04717505110430014358531778979, 7.49070531765642687188640723413, 8.313019769084536360193690639892, 9.893982389106184268672414672747, 10.62509361943127686838005164105, 11.92941000245895948639191161844
