| L(s) = 1 | + 2-s + 3.04·3-s + 4-s − 0.864·5-s + 3.04·6-s + 1.15·7-s + 8-s + 6.27·9-s − 0.864·10-s + 3.58·11-s + 3.04·12-s − 4.12·13-s + 1.15·14-s − 2.63·15-s + 16-s + 4.56·17-s + 6.27·18-s − 2.59·19-s − 0.864·20-s + 3.51·21-s + 3.58·22-s − 0.859·23-s + 3.04·24-s − 4.25·25-s − 4.12·26-s + 9.96·27-s + 1.15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.386·5-s + 1.24·6-s + 0.436·7-s + 0.353·8-s + 2.09·9-s − 0.273·10-s + 1.08·11-s + 0.879·12-s − 1.14·13-s + 0.308·14-s − 0.679·15-s + 0.250·16-s + 1.10·17-s + 1.47·18-s − 0.594·19-s − 0.193·20-s + 0.766·21-s + 0.764·22-s − 0.179·23-s + 0.621·24-s − 0.850·25-s − 0.808·26-s + 1.91·27-s + 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.659800241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.659800241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 0.864T + 5T^{2} \) |
| 7 | \( 1 - 1.15T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 2.59T + 19T^{2} \) |
| 23 | \( 1 + 0.859T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 + 2.98T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 7.08T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 1.49T + 79T^{2} \) |
| 83 | \( 1 - 8.41T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 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
−8.076982728262326737820977125263, −7.46702477327917115560130585144, −6.88324235879555420263923887083, −5.97388201075133872457461731322, −4.85720265672549407807460469243, −4.22679344993764598768302135698, −3.67161386604690210633428452764, −2.85504656598549514096174071101, −2.18015550205187021063125407656, −1.24444494542067995295559580369,
1.24444494542067995295559580369, 2.18015550205187021063125407656, 2.85504656598549514096174071101, 3.67161386604690210633428452764, 4.22679344993764598768302135698, 4.85720265672549407807460469243, 5.97388201075133872457461731322, 6.88324235879555420263923887083, 7.46702477327917115560130585144, 8.076982728262326737820977125263
