| L(s) = 1 | + 2-s − 3.13·3-s + 4-s − 3.90·5-s − 3.13·6-s − 7-s + 8-s + 6.84·9-s − 3.90·10-s − 6.13·11-s − 3.13·12-s − 2.06·13-s − 14-s + 12.2·15-s + 16-s − 0.890·17-s + 6.84·18-s − 0.719·19-s − 3.90·20-s + 3.13·21-s − 6.13·22-s − 3.13·24-s + 10.2·25-s − 2.06·26-s − 12.0·27-s − 28-s + 5.88·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.81·3-s + 0.5·4-s − 1.74·5-s − 1.28·6-s − 0.377·7-s + 0.353·8-s + 2.28·9-s − 1.23·10-s − 1.85·11-s − 0.905·12-s − 0.573·13-s − 0.267·14-s + 3.16·15-s + 0.250·16-s − 0.216·17-s + 1.61·18-s − 0.165·19-s − 0.872·20-s + 0.684·21-s − 1.30·22-s − 0.640·24-s + 2.04·25-s − 0.405·26-s − 2.32·27-s − 0.188·28-s + 1.09·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 + 3.90T + 5T^{2} \) |
| 11 | \( 1 + 6.13T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 + 0.890T + 17T^{2} \) |
| 19 | \( 1 + 0.719T + 19T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 - 6.18T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 6.59T + 61T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
| 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
−7.39002929744431607630625049624, −6.81816866657474712486497616659, −6.03363109479205469390653122628, −5.32111555816975887197514534258, −4.68896848380574112020725030941, −4.34620525561960837071960713759, −3.34361748455180359880207710617, −2.45908323389126442838414061225, −0.77560549393060987822880403083, 0,
0.77560549393060987822880403083, 2.45908323389126442838414061225, 3.34361748455180359880207710617, 4.34620525561960837071960713759, 4.68896848380574112020725030941, 5.32111555816975887197514534258, 6.03363109479205469390653122628, 6.81816866657474712486497616659, 7.39002929744431607630625049624
