| L(s) = 1 | − 2-s + 4-s + 5-s + 1.27·7-s − 8-s − 10-s − 1.54·11-s + 5.33·13-s − 1.27·14-s + 16-s − 6.54·17-s − 4.60·19-s + 20-s + 1.54·22-s − 2.65·23-s + 25-s − 5.33·26-s + 1.27·28-s − 1.90·29-s + 6.46·31-s − 32-s + 6.54·34-s + 1.27·35-s − 2.38·37-s + 4.60·38-s − 40-s + 1.21·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.483·7-s − 0.353·8-s − 0.316·10-s − 0.466·11-s + 1.47·13-s − 0.342·14-s + 0.250·16-s − 1.58·17-s − 1.05·19-s + 0.223·20-s + 0.329·22-s − 0.553·23-s + 0.200·25-s − 1.04·26-s + 0.241·28-s − 0.353·29-s + 1.16·31-s − 0.176·32-s + 1.12·34-s + 0.216·35-s − 0.391·37-s + 0.746·38-s − 0.158·40-s + 0.189·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 + 2.65T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 2.38T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 + 0.339T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 6.64T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 + 4.88T + 67T^{2} \) |
| 71 | \( 1 - 1.30T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 2.09T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 97 | \( 1 - 7.01T + 97T^{2} \) |
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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
−7.66196124545305457332781298404, −6.69056860333227958489202260235, −6.28287819077571626993469051929, −5.60795615118845650760977063871, −4.58765210791530686546402333909, −3.98156775820822839656575482688, −2.82656620323708544517545496853, −2.07117715300573499246640499904, −1.31415267473386030018287659366, 0,
1.31415267473386030018287659366, 2.07117715300573499246640499904, 2.82656620323708544517545496853, 3.98156775820822839656575482688, 4.58765210791530686546402333909, 5.60795615118845650760977063871, 6.28287819077571626993469051929, 6.69056860333227958489202260235, 7.66196124545305457332781298404
