| L(s) = 1 | − 2.22·2-s + 2.93·4-s − 0.179·5-s − 5.01·7-s − 2.06·8-s + 0.399·10-s − 11-s + 2.16·13-s + 11.1·14-s − 1.26·16-s − 4.79·17-s + 6.94·19-s − 0.527·20-s + 2.22·22-s + 1.92·23-s − 4.96·25-s − 4.81·26-s − 14.7·28-s + 4.05·29-s − 6.04·31-s + 6.95·32-s + 10.6·34-s + 0.902·35-s + 10.1·37-s − 15.4·38-s + 0.372·40-s − 8.33·41-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.46·4-s − 0.0804·5-s − 1.89·7-s − 0.731·8-s + 0.126·10-s − 0.301·11-s + 0.601·13-s + 2.97·14-s − 0.317·16-s − 1.16·17-s + 1.59·19-s − 0.117·20-s + 0.473·22-s + 0.402·23-s − 0.993·25-s − 0.943·26-s − 2.77·28-s + 0.752·29-s − 1.08·31-s + 1.22·32-s + 1.82·34-s + 0.152·35-s + 1.66·37-s − 2.50·38-s + 0.0588·40-s − 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 + 0.179T + 5T^{2} \) |
| 7 | \( 1 + 5.01T + 7T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 - 1.92T + 23T^{2} \) |
| 29 | \( 1 - 4.05T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 8.33T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 2.10T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 5.82T + 89T^{2} \) |
| 97 | \( 1 - 0.569T + 97T^{2} \) |
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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
−7.75914164914854140454356614992, −7.20031682249263986338808834151, −6.46404301866778201698478548207, −6.05478443931407276540386296198, −4.89454956267118952440875953220, −3.72230080570098442668281000335, −3.04380990903886338035412386558, −2.15348888165016116399230410254, −0.915566482065624920702600020297, 0,
0.915566482065624920702600020297, 2.15348888165016116399230410254, 3.04380990903886338035412386558, 3.72230080570098442668281000335, 4.89454956267118952440875953220, 6.05478443931407276540386296198, 6.46404301866778201698478548207, 7.20031682249263986338808834151, 7.75914164914854140454356614992
