| L(s) = 1 | + 0.498·2-s − 3-s − 1.75·4-s + 0.622·5-s − 0.498·6-s + 0.110·7-s − 1.87·8-s + 9-s + 0.310·10-s + 0.189·11-s + 1.75·12-s + 6.61·13-s + 0.0548·14-s − 0.622·15-s + 2.56·16-s − 17-s + 0.498·18-s − 0.0616·19-s − 1.09·20-s − 0.110·21-s + 0.0947·22-s − 5.46·23-s + 1.87·24-s − 4.61·25-s + 3.30·26-s − 27-s − 0.192·28-s + ⋯ |
| L(s) = 1 | + 0.352·2-s − 0.577·3-s − 0.875·4-s + 0.278·5-s − 0.203·6-s + 0.0415·7-s − 0.661·8-s + 0.333·9-s + 0.0982·10-s + 0.0572·11-s + 0.505·12-s + 1.83·13-s + 0.0146·14-s − 0.160·15-s + 0.642·16-s − 0.242·17-s + 0.117·18-s − 0.0141·19-s − 0.243·20-s − 0.0240·21-s + 0.0201·22-s − 1.14·23-s + 0.382·24-s − 0.922·25-s + 0.647·26-s − 0.192·27-s − 0.0364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 0.498T + 2T^{2} \) |
| 5 | \( 1 - 0.622T + 5T^{2} \) |
| 7 | \( 1 - 0.110T + 7T^{2} \) |
| 11 | \( 1 - 0.189T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 19 | \( 1 + 0.0616T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 4.08T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 0.0568T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−7.57029367367483568582152432692, −6.43691612081093638026216001262, −5.99005226092682241596752060050, −5.52212975777940813730918415678, −4.66759875437916479090641008567, −3.83899829606638902072553781392, −3.56327881395926606404629966050, −2.12102829379722973887988940137, −1.16459774122004603678270100432, 0,
1.16459774122004603678270100432, 2.12102829379722973887988940137, 3.56327881395926606404629966050, 3.83899829606638902072553781392, 4.66759875437916479090641008567, 5.52212975777940813730918415678, 5.99005226092682241596752060050, 6.43691612081093638026216001262, 7.57029367367483568582152432692
