| L(s) = 1 | − 2-s − 0.896·3-s + 4-s − 0.517·5-s + 0.896·6-s − 8-s − 2.19·9-s + 0.517·10-s + 0.464·11-s − 0.896·12-s − 5.79·13-s + 0.464·15-s + 16-s + 5.65·17-s + 2.19·18-s + 5.27·19-s − 0.517·20-s − 0.464·22-s + 2·23-s + 0.896·24-s − 4.73·25-s + 5.79·26-s + 4.65·27-s − 29-s − 0.464·30-s + 8.24·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.517·3-s + 0.5·4-s − 0.231·5-s + 0.366·6-s − 0.353·8-s − 0.732·9-s + 0.163·10-s + 0.139·11-s − 0.258·12-s − 1.60·13-s + 0.119·15-s + 0.250·16-s + 1.37·17-s + 0.517·18-s + 1.21·19-s − 0.115·20-s − 0.0989·22-s + 0.417·23-s + 0.183·24-s − 0.946·25-s + 1.13·26-s + 0.896·27-s − 0.185·29-s − 0.0847·30-s + 1.48·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.896T + 3T^{2} \) |
| 5 | \( 1 + 0.517T + 5T^{2} \) |
| 11 | \( 1 - 0.464T + 11T^{2} \) |
| 13 | \( 1 + 5.79T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 0.757T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.283535270565763116783365137689, −7.70051571961644093014164936373, −7.09494578755350914979560003840, −6.11405823134162359480203476711, −5.41241779859835878049553594903, −4.68939974954955118796342984812, −3.30994597794425797223401039226, −2.62171885007143553708471718568, −1.20730331634176497959639156771, 0,
1.20730331634176497959639156771, 2.62171885007143553708471718568, 3.30994597794425797223401039226, 4.68939974954955118796342984812, 5.41241779859835878049553594903, 6.11405823134162359480203476711, 7.09494578755350914979560003840, 7.70051571961644093014164936373, 8.283535270565763116783365137689
