L(s) = 1 | + 2-s − 1.35·3-s + 4-s + 1.14·5-s − 1.35·6-s − 7-s + 8-s − 1.16·9-s + 1.14·10-s − 4.90·11-s − 1.35·12-s + 3.53·13-s − 14-s − 1.55·15-s + 16-s − 2.10·17-s − 1.16·18-s + 2.64·19-s + 1.14·20-s + 1.35·21-s − 4.90·22-s + 1.73·23-s − 1.35·24-s − 3.68·25-s + 3.53·26-s + 5.64·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.782·3-s + 0.5·4-s + 0.512·5-s − 0.553·6-s − 0.377·7-s + 0.353·8-s − 0.386·9-s + 0.362·10-s − 1.47·11-s − 0.391·12-s + 0.980·13-s − 0.267·14-s − 0.401·15-s + 0.250·16-s − 0.510·17-s − 0.273·18-s + 0.605·19-s + 0.256·20-s + 0.295·21-s − 1.04·22-s + 0.360·23-s − 0.276·24-s − 0.737·25-s + 0.693·26-s + 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 5 | \( 1 - 1.14T + 5T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 3.53T + 59T^{2} \) |
| 61 | \( 1 + 0.0973T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 + 5.06T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 + 7.91T + 83T^{2} \) |
| 89 | \( 1 - 0.879T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.66830272495015724487201640938, −6.68630688472197087639808483749, −6.05199389288550617609441929792, −5.69220564554747261051347489350, −4.97161387144551800877659823884, −4.24437532148533857197564927970, −3.06246449113217523307091358402, −2.62750223451303194719828844997, −1.34800836362455330124231726181, 0,
1.34800836362455330124231726181, 2.62750223451303194719828844997, 3.06246449113217523307091358402, 4.24437532148533857197564927970, 4.97161387144551800877659823884, 5.69220564554747261051347489350, 6.05199389288550617609441929792, 6.68630688472197087639808483749, 7.66830272495015724487201640938