| L(s) = 1 | − 0.508·2-s + 2.08·3-s − 1.74·4-s + 5-s − 1.05·6-s − 7-s + 1.90·8-s + 1.35·9-s − 0.508·10-s + 0.775·11-s − 3.63·12-s − 3.72·13-s + 0.508·14-s + 2.08·15-s + 2.51·16-s − 4.85·17-s − 0.686·18-s + 1.98·19-s − 1.74·20-s − 2.08·21-s − 0.394·22-s + 3.04·23-s + 3.96·24-s + 25-s + 1.89·26-s − 3.43·27-s + 1.74·28-s + ⋯ |
| L(s) = 1 | − 0.359·2-s + 1.20·3-s − 0.870·4-s + 0.447·5-s − 0.432·6-s − 0.377·7-s + 0.672·8-s + 0.450·9-s − 0.160·10-s + 0.233·11-s − 1.04·12-s − 1.03·13-s + 0.135·14-s + 0.538·15-s + 0.629·16-s − 1.17·17-s − 0.161·18-s + 0.456·19-s − 0.389·20-s − 0.455·21-s − 0.0840·22-s + 0.634·23-s + 0.809·24-s + 0.200·25-s + 0.371·26-s − 0.661·27-s + 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 0.508T + 2T^{2} \) |
| 3 | \( 1 - 2.08T + 3T^{2} \) |
| 11 | \( 1 - 0.775T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 + 8.76T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 3.57T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 53 | \( 1 - 7.87T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 0.864T + 61T^{2} \) |
| 67 | \( 1 + 7.24T + 67T^{2} \) |
| 71 | \( 1 + 8.33T + 71T^{2} \) |
| 73 | \( 1 - 1.08T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + 0.886T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−7.55192323488858528826176246510, −7.10414050579997945720061449294, −6.16691579822876188805296703565, −5.24756679539709696586332034397, −4.57515825881047998223720298317, −3.86330666708736600962483335834, −2.94703698772473343547329986005, −2.38108868184102852383832712575, −1.30925618551172256629528795324, 0,
1.30925618551172256629528795324, 2.38108868184102852383832712575, 2.94703698772473343547329986005, 3.86330666708736600962483335834, 4.57515825881047998223720298317, 5.24756679539709696586332034397, 6.16691579822876188805296703565, 7.10414050579997945720061449294, 7.55192323488858528826176246510
