| L(s) = 1 | − 2-s + 0.328·3-s + 4-s + 5-s − 0.328·6-s − 2.28·7-s − 8-s − 2.89·9-s − 10-s + 11-s + 0.328·12-s − 1.97·13-s + 2.28·14-s + 0.328·15-s + 16-s − 2.34·17-s + 2.89·18-s + 6.58·19-s + 20-s − 0.748·21-s − 22-s + 1.49·23-s − 0.328·24-s + 25-s + 1.97·26-s − 1.93·27-s − 2.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.189·3-s + 0.5·4-s + 0.447·5-s − 0.133·6-s − 0.862·7-s − 0.353·8-s − 0.964·9-s − 0.316·10-s + 0.301·11-s + 0.0947·12-s − 0.547·13-s + 0.609·14-s + 0.0847·15-s + 0.250·16-s − 0.569·17-s + 0.681·18-s + 1.51·19-s + 0.223·20-s − 0.163·21-s − 0.213·22-s + 0.312·23-s − 0.0669·24-s + 0.200·25-s + 0.386·26-s − 0.372·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 0.328T + 3T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 0.561T + 37T^{2} \) |
| 41 | \( 1 - 2.80T + 41T^{2} \) |
| 43 | \( 1 + 3.31T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 - 2.99T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 - 8.15T + 71T^{2} \) |
| 79 | \( 1 + 6.79T + 79T^{2} \) |
| 83 | \( 1 - 5.33T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 6.86T + 97T^{2} \) |
| 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
−7.57606307540176524035467659191, −6.73206360434508633626952954077, −6.34207791331799181720338313865, −5.49798747439438000137401067289, −4.84022734159009603092011842658, −3.56704236956857294497129531550, −2.95967930441300244953430728062, −2.29935814926219356198393769938, −1.13548464259047681965481674227, 0,
1.13548464259047681965481674227, 2.29935814926219356198393769938, 2.95967930441300244953430728062, 3.56704236956857294497129531550, 4.84022734159009603092011842658, 5.49798747439438000137401067289, 6.34207791331799181720338313865, 6.73206360434508633626952954077, 7.57606307540176524035467659191
