L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 2.67·5-s − 1.61·6-s − 0.794·7-s − 8-s − 0.391·9-s + 2.67·10-s − 2.37·11-s + 1.61·12-s − 0.106·13-s + 0.794·14-s − 4.32·15-s + 16-s + 2.83·17-s + 0.391·18-s − 19-s − 2.67·20-s − 1.28·21-s + 2.37·22-s + 6.31·23-s − 1.61·24-s + 2.16·25-s + 0.106·26-s − 5.47·27-s − 0.794·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.932·3-s + 0.5·4-s − 1.19·5-s − 0.659·6-s − 0.300·7-s − 0.353·8-s − 0.130·9-s + 0.846·10-s − 0.714·11-s + 0.466·12-s − 0.0294·13-s + 0.212·14-s − 1.11·15-s + 0.250·16-s + 0.688·17-s + 0.0923·18-s − 0.229·19-s − 0.598·20-s − 0.279·21-s + 0.505·22-s + 1.31·23-s − 0.329·24-s + 0.432·25-s + 0.0208·26-s − 1.05·27-s − 0.150·28-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 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 + 0.794T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 + 0.106T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 1.48T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 - 8.75T + 37T^{2} \) |
| 41 | \( 1 + 9.87T + 41T^{2} \) |
| 43 | \( 1 - 0.390T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 + 7.41T + 61T^{2} \) |
| 67 | \( 1 + 7.63T + 67T^{2} \) |
| 71 | \( 1 - 5.42T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 3.26T + 89T^{2} \) |
| 97 | \( 1 + 9.02T + 97T^{2} \) |
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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
−7.81804642227327553581212452024, −7.10537044504153298168582023309, −6.34728502982198250648838528294, −5.39651871307871485475005946019, −4.52431852454437651233180345532, −3.61454305458808009052602252078, −2.99965415142208800589618264216, −2.45478797058236335170889221052, −1.09636677486236666189832742111, 0,
1.09636677486236666189832742111, 2.45478797058236335170889221052, 2.99965415142208800589618264216, 3.61454305458808009052602252078, 4.52431852454437651233180345532, 5.39651871307871485475005946019, 6.34728502982198250648838528294, 7.10537044504153298168582023309, 7.81804642227327553581212452024