| L(s) = 1 | + 2.16·2-s + 3-s + 2.70·4-s + 1.45·5-s + 2.16·6-s + 1.53·8-s + 9-s + 3.15·10-s − 5.83·11-s + 2.70·12-s − 6.73·13-s + 1.45·15-s − 2.08·16-s + 0.451·17-s + 2.16·18-s − 6.09·19-s + 3.93·20-s − 12.6·22-s + 1.51·23-s + 1.53·24-s − 2.89·25-s − 14.6·26-s + 27-s + 1.27·29-s + 3.15·30-s + 3.08·31-s − 7.59·32-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.577·3-s + 1.35·4-s + 0.649·5-s + 0.885·6-s + 0.542·8-s + 0.333·9-s + 0.996·10-s − 1.75·11-s + 0.781·12-s − 1.86·13-s + 0.375·15-s − 0.521·16-s + 0.109·17-s + 0.511·18-s − 1.39·19-s + 0.879·20-s − 2.69·22-s + 0.316·23-s + 0.312·24-s − 0.578·25-s − 2.86·26-s + 0.192·27-s + 0.237·29-s + 0.575·30-s + 0.554·31-s − 1.34·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 11 | \( 1 + 5.83T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 - 0.451T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 8.05T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 + 0.0382T + 61T^{2} \) |
| 67 | \( 1 + 0.886T + 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 - 5.91T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 - 2.01T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
| 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
−7.51620255373048542700890698778, −6.71516574186619173686077617965, −5.93208066828803723693581733789, −5.31977313777083821016571258740, −4.70221157011552292662217114223, −4.17926438934885466109141507979, −2.95850837080727030550589511078, −2.55944130538555225900635215639, −2.02368775868718601775069331508, 0,
2.02368775868718601775069331508, 2.55944130538555225900635215639, 2.95850837080727030550589511078, 4.17926438934885466109141507979, 4.70221157011552292662217114223, 5.31977313777083821016571258740, 5.93208066828803723693581733789, 6.71516574186619173686077617965, 7.51620255373048542700890698778
