L(s) = 1 | − 1.54·2-s + 0.389·4-s + 0.315·5-s + 2.48·8-s − 0.487·10-s + 1.80·11-s − 5.89·13-s − 4.62·16-s + 4.31·17-s + 6.30·19-s + 0.122·20-s − 2.79·22-s + 6.44·23-s − 4.90·25-s + 9.11·26-s − 7.97·29-s + 5.19·31-s + 2.17·32-s − 6.66·34-s − 1.41·37-s − 9.73·38-s + 0.785·40-s − 2.17·41-s + 1.04·43-s + 0.704·44-s − 9.95·46-s + 1.17·47-s + ⋯ |
L(s) = 1 | − 1.09·2-s + 0.194·4-s + 0.141·5-s + 0.880·8-s − 0.154·10-s + 0.545·11-s − 1.63·13-s − 1.15·16-s + 1.04·17-s + 1.44·19-s + 0.0274·20-s − 0.596·22-s + 1.34·23-s − 0.980·25-s + 1.78·26-s − 1.48·29-s + 0.932·31-s + 0.383·32-s − 1.14·34-s − 0.232·37-s − 1.57·38-s + 0.124·40-s − 0.339·41-s + 0.159·43-s + 0.106·44-s − 1.46·46-s + 0.171·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9241490029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9241490029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 5 | \( 1 - 0.315T + 5T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 - 8.37T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + 8.12T + 73T^{2} \) |
| 79 | \( 1 + 1.13T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 9.25T + 97T^{2} \) |
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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
−8.892362512562076551089832271956, −7.85252925810450745641428351694, −7.46339966405303174484247038689, −6.83193653383949477418062825847, −5.52348839582633745955340882267, −5.01556620699006324369506550677, −3.95237012227271368609584502099, −2.89009909211126518119566394487, −1.74060218904714309776079994374, −0.71327219685320034485140629839,
0.71327219685320034485140629839, 1.74060218904714309776079994374, 2.89009909211126518119566394487, 3.95237012227271368609584502099, 5.01556620699006324369506550677, 5.52348839582633745955340882267, 6.83193653383949477418062825847, 7.46339966405303174484247038689, 7.85252925810450745641428351694, 8.892362512562076551089832271956