| L(s) = 1 | − 2.33·2-s + 3.44·4-s − 2·7-s − 3.38·8-s − 2·13-s + 4.66·14-s + 1.00·16-s − 1.04·17-s − 2.89·19-s + 5.71·23-s + 4.66·26-s − 6.89·28-s − 8.28·29-s + 6.89·31-s + 4.43·32-s + 2.44·34-s − 37-s + 6.76·38-s + 2.09·41-s − 6.89·43-s − 13.3·46-s + 11.4·47-s − 3·49-s − 6.89·52-s − 4.19·53-s + 6.76·56-s + 19.3·58-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.72·4-s − 0.755·7-s − 1.19·8-s − 0.554·13-s + 1.24·14-s + 0.250·16-s − 0.254·17-s − 0.665·19-s + 1.19·23-s + 0.915·26-s − 1.30·28-s − 1.53·29-s + 1.23·31-s + 0.783·32-s + 0.420·34-s − 0.164·37-s + 1.09·38-s + 0.327·41-s − 1.05·43-s − 1.96·46-s + 1.66·47-s − 0.428·49-s − 0.956·52-s − 0.576·53-s + 0.904·56-s + 2.54·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 - 6.89T + 31T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.19T + 53T^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 + 0.898T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 7.23T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
| 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
−7.56799696884915141497272029536, −6.89732708067097493264041977795, −6.52493158425753899016148384360, −5.58594697753587214437034688859, −4.67396788760632657786120562474, −3.68297521161163241584710436865, −2.71401241908904343176798365649, −2.06298875650410320180190594781, −0.945113581653782060312055055327, 0,
0.945113581653782060312055055327, 2.06298875650410320180190594781, 2.71401241908904343176798365649, 3.68297521161163241584710436865, 4.67396788760632657786120562474, 5.58594697753587214437034688859, 6.52493158425753899016148384360, 6.89732708067097493264041977795, 7.56799696884915141497272029536
