| L(s) = 1 | − 1.33·2-s + 0.790·4-s + 1.95·5-s + 0.279·8-s − 2.61·10-s − 1.82·11-s − 1.16·16-s + 17-s + 1.54·20-s + 2.44·22-s + 2.82·25-s + 0.209·29-s + 1.33·31-s + 1.27·32-s − 1.33·34-s + 0.547·40-s − 1.44·44-s + 49-s − 3.78·50-s − 3.57·55-s − 0.279·58-s − 61-s − 1.79·62-s − 0.547·64-s − 67-s + 0.790·68-s + 1.61·71-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.790·4-s + 1.95·5-s + 0.279·8-s − 2.61·10-s − 1.82·11-s − 1.16·16-s + 17-s + 1.54·20-s + 2.44·22-s + 2.82·25-s + 0.209·29-s + 1.33·31-s + 1.27·32-s − 1.33·34-s + 0.547·40-s − 1.44·44-s + 49-s − 3.78·50-s − 3.57·55-s − 0.279·58-s − 61-s − 1.79·62-s − 0.547·64-s − 67-s + 0.790·68-s + 1.61·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8002180020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8002180020\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + T^{2} \) |
| 5 | \( 1 - 1.95T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.82T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.209T + T^{2} \) |
| 31 | \( 1 - 1.33T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.209T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.470591870397296430003560227192, −8.568716568564762653917468675905, −7.953972416997392909019611840794, −7.12517713367536821432515400006, −6.18451489857585031771027563507, −5.43770612561425565819200350031, −4.76622640918086845544745599998, −2.87275933253349990916242086529, −2.21356132230458549059499580294, −1.13145492751866046510513353819,
1.13145492751866046510513353819, 2.21356132230458549059499580294, 2.87275933253349990916242086529, 4.76622640918086845544745599998, 5.43770612561425565819200350031, 6.18451489857585031771027563507, 7.12517713367536821432515400006, 7.953972416997392909019611840794, 8.568716568564762653917468675905, 9.470591870397296430003560227192
