| L(s) = 1 | + 2-s + 2.74·3-s + 4-s − 2.12·5-s + 2.74·6-s − 0.877·7-s + 8-s + 4.52·9-s − 2.12·10-s − 11-s + 2.74·12-s − 3.39·13-s − 0.877·14-s − 5.83·15-s + 16-s − 3.03·17-s + 4.52·18-s − 2.12·20-s − 2.40·21-s − 22-s + 2.43·23-s + 2.74·24-s − 0.480·25-s − 3.39·26-s + 4.19·27-s − 0.877·28-s − 3.95·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.58·3-s + 0.5·4-s − 0.950·5-s + 1.12·6-s − 0.331·7-s + 0.353·8-s + 1.50·9-s − 0.672·10-s − 0.301·11-s + 0.792·12-s − 0.940·13-s − 0.234·14-s − 1.50·15-s + 0.250·16-s − 0.736·17-s + 1.06·18-s − 0.475·20-s − 0.525·21-s − 0.213·22-s + 0.507·23-s + 0.560·24-s − 0.0960·25-s − 0.665·26-s + 0.807·27-s − 0.165·28-s − 0.734·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 + 0.877T + 7T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 - 2.25T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 5.36T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 - 8.88T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.61581159607999819856800477503, −7.05905163726895448807439076283, −6.20668096508086571044011233962, −5.11367587379466360006425926997, −4.46548036511044007825389248739, −3.73800427022498362912151208332, −3.21785953698397577767453393550, −2.50723317184675712955960883712, −1.74892997219920836412079456263, 0,
1.74892997219920836412079456263, 2.50723317184675712955960883712, 3.21785953698397577767453393550, 3.73800427022498362912151208332, 4.46548036511044007825389248739, 5.11367587379466360006425926997, 6.20668096508086571044011233962, 7.05905163726895448807439076283, 7.61581159607999819856800477503
