| L(s) = 1 | + 2.45·2-s + 2.80·3-s + 4.03·4-s − 3.48·5-s + 6.87·6-s − 3.54·7-s + 4.99·8-s + 4.84·9-s − 8.56·10-s − 5.60·11-s + 11.2·12-s − 8.69·14-s − 9.76·15-s + 4.20·16-s − 2.47·17-s + 11.8·18-s − 2.94·19-s − 14.0·20-s − 9.91·21-s − 13.7·22-s + 1.33·23-s + 13.9·24-s + 7.14·25-s + 5.15·27-s − 14.2·28-s − 6.22·29-s − 23.9·30-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.61·3-s + 2.01·4-s − 1.55·5-s + 2.80·6-s − 1.33·7-s + 1.76·8-s + 1.61·9-s − 2.70·10-s − 1.69·11-s + 3.26·12-s − 2.32·14-s − 2.52·15-s + 1.05·16-s − 0.601·17-s + 2.80·18-s − 0.675·19-s − 3.14·20-s − 2.16·21-s − 2.93·22-s + 0.277·23-s + 2.85·24-s + 1.42·25-s + 0.992·27-s − 2.69·28-s − 1.15·29-s − 4.37·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 - 2.80T + 3T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 - 1.33T + 23T^{2} \) |
| 29 | \( 1 + 6.22T + 29T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 + 0.926T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 - 5.72T + 61T^{2} \) |
| 67 | \( 1 - 9.27T + 67T^{2} \) |
| 71 | \( 1 + 0.850T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.72T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69343646349794477765024139484, −7.17102435915771083101289253048, −6.46630315852488747803273222998, −5.46329469094129373316481145923, −4.50450504480051358982106377855, −3.96832770541835026452007441058, −3.34704816008969175933973191828, −2.83342225481433173519312302583, −2.24532660892996888005665762445, 0,
2.24532660892996888005665762445, 2.83342225481433173519312302583, 3.34704816008969175933973191828, 3.96832770541835026452007441058, 4.50450504480051358982106377855, 5.46329469094129373316481145923, 6.46630315852488747803273222998, 7.17102435915771083101289253048, 7.69343646349794477765024139484
