L(s) = 1 | + 2.42·2-s + 3-s + 3.86·4-s − 4.19·5-s + 2.42·6-s − 0.311·7-s + 4.51·8-s + 9-s − 10.1·10-s + 1.14·11-s + 3.86·12-s − 0.0687·13-s − 0.754·14-s − 4.19·15-s + 3.20·16-s + 17-s + 2.42·18-s − 3.46·19-s − 16.2·20-s − 0.311·21-s + 2.78·22-s − 6.85·23-s + 4.51·24-s + 12.5·25-s − 0.166·26-s + 27-s − 1.20·28-s + ⋯ |
L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.93·4-s − 1.87·5-s + 0.988·6-s − 0.117·7-s + 1.59·8-s + 0.333·9-s − 3.21·10-s + 0.346·11-s + 1.11·12-s − 0.0190·13-s − 0.201·14-s − 1.08·15-s + 0.800·16-s + 0.242·17-s + 0.570·18-s − 0.795·19-s − 3.62·20-s − 0.0679·21-s + 0.593·22-s − 1.42·23-s + 0.921·24-s + 2.51·25-s − 0.0326·26-s + 0.192·27-s − 0.227·28-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 | \( 1 - 2.42T + 2T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 7 | \( 1 + 0.311T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 + 0.0687T + 13T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 + 0.351T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 + 2.33T + 61T^{2} \) |
| 67 | \( 1 + 9.21T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.33999699265172095789777580063, −6.72250959352983208736985853145, −6.11498614355061912369487720527, −5.07461419087478756097654000771, −4.35612827913518859890724532546, −4.01985787327720899235971117403, −3.35559479866607465295453049343, −2.77864077553709742213793735871, −1.65122103842030231688995674902, 0,
1.65122103842030231688995674902, 2.77864077553709742213793735871, 3.35559479866607465295453049343, 4.01985787327720899235971117403, 4.35612827913518859890724532546, 5.07461419087478756097654000771, 6.11498614355061912369487720527, 6.72250959352983208736985853145, 7.33999699265172095789777580063