L(s) = 1 | − 2-s + 2.33·3-s + 4-s − 3.15·5-s − 2.33·6-s + 3.24·7-s − 8-s + 2.44·9-s + 3.15·10-s + 3.17·11-s + 2.33·12-s − 6.52·13-s − 3.24·14-s − 7.35·15-s + 16-s − 2.42·17-s − 2.44·18-s + 19-s − 3.15·20-s + 7.56·21-s − 3.17·22-s + 0.984·23-s − 2.33·24-s + 4.93·25-s + 6.52·26-s − 1.29·27-s + 3.24·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s − 1.40·5-s − 0.952·6-s + 1.22·7-s − 0.353·8-s + 0.814·9-s + 0.996·10-s + 0.955·11-s + 0.673·12-s − 1.80·13-s − 0.866·14-s − 1.89·15-s + 0.250·16-s − 0.588·17-s − 0.576·18-s + 0.229·19-s − 0.704·20-s + 1.65·21-s − 0.675·22-s + 0.205·23-s − 0.476·24-s + 0.987·25-s + 1.27·26-s − 0.249·27-s + 0.612·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 23 | \( 1 - 0.984T + 23T^{2} \) |
| 29 | \( 1 - 6.16T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 + 0.623T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 0.804T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 0.328T + 83T^{2} \) |
| 89 | \( 1 - 0.785T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.70089344546357559306303743964, −7.25978877291927049446996535966, −6.53733070163912070631755048602, −5.09048996793543174406515225852, −4.52421552190527289076567868791, −3.79810728612999396113823900029, −2.98204065776885017249077711299, −2.23334805197512201985210954926, −1.36585798897800293977272610461, 0,
1.36585798897800293977272610461, 2.23334805197512201985210954926, 2.98204065776885017249077711299, 3.79810728612999396113823900029, 4.52421552190527289076567868791, 5.09048996793543174406515225852, 6.53733070163912070631755048602, 7.25978877291927049446996535966, 7.70089344546357559306303743964