| L(s) = 1 | − 2-s − 2.23·3-s + 4-s − 2.71·5-s + 2.23·6-s − 8-s + 2.01·9-s + 2.71·10-s + 3.73·11-s − 2.23·12-s − 3.06·13-s + 6.09·15-s + 16-s − 5.82·17-s − 2.01·18-s + 1.54·19-s − 2.71·20-s − 3.73·22-s + 2.39·23-s + 2.23·24-s + 2.39·25-s + 3.06·26-s + 2.20·27-s − 29-s − 6.09·30-s − 3.41·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s − 1.21·5-s + 0.914·6-s − 0.353·8-s + 0.672·9-s + 0.859·10-s + 1.12·11-s − 0.646·12-s − 0.849·13-s + 1.57·15-s + 0.250·16-s − 1.41·17-s − 0.475·18-s + 0.354·19-s − 0.607·20-s − 0.796·22-s + 0.499·23-s + 0.457·24-s + 0.478·25-s + 0.600·26-s + 0.423·27-s − 0.185·29-s − 1.11·30-s − 0.612·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 + 2.71T + 5T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 - 5.61T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 9.71T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 0.150T + 97T^{2} \) |
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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
−8.405924405856592046082525680459, −7.53769906585950459688224368595, −6.90264231436436918320433287677, −6.36519085276343438738576655538, −5.36210402769674996931154809618, −4.49056357697204650258582798899, −3.77773793262733775569761663638, −2.43606853080874667852167528849, −0.969615414519871914200733060874, 0,
0.969615414519871914200733060874, 2.43606853080874667852167528849, 3.77773793262733775569761663638, 4.49056357697204650258582798899, 5.36210402769674996931154809618, 6.36519085276343438738576655538, 6.90264231436436918320433287677, 7.53769906585950459688224368595, 8.405924405856592046082525680459
