| L(s) = 1 | − 2.70·2-s − 3-s + 5.32·4-s + 2.36·5-s + 2.70·6-s + 2.88·7-s − 9.00·8-s + 9-s − 6.41·10-s + 1.43·11-s − 5.32·12-s + 13-s − 7.80·14-s − 2.36·15-s + 13.7·16-s − 6.36·17-s − 2.70·18-s + 4.50·19-s + 12.6·20-s − 2.88·21-s − 3.89·22-s − 8.58·23-s + 9.00·24-s + 0.611·25-s − 2.70·26-s − 27-s + 15.3·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.577·3-s + 2.66·4-s + 1.05·5-s + 1.10·6-s + 1.08·7-s − 3.18·8-s + 0.333·9-s − 2.02·10-s + 0.434·11-s − 1.53·12-s + 0.277·13-s − 2.08·14-s − 0.611·15-s + 3.43·16-s − 1.54·17-s − 0.638·18-s + 1.03·19-s + 2.82·20-s − 0.628·21-s − 0.830·22-s − 1.78·23-s + 1.83·24-s + 0.122·25-s − 0.530·26-s − 0.192·27-s + 2.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 4.50T + 19T^{2} \) |
| 23 | \( 1 + 8.58T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 3.66T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 - 8.54T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.212973313819630460444369298086, −7.56241282060100871968041344083, −6.63906349572675573240747739258, −6.23732539081062502110840150965, −5.43504156820550579287645552794, −4.38390014051080959790306978480, −2.86181110195019404152402843296, −1.66954965294014473855387262608, −1.58845672706565736209810024570, 0,
1.58845672706565736209810024570, 1.66954965294014473855387262608, 2.86181110195019404152402843296, 4.38390014051080959790306978480, 5.43504156820550579287645552794, 6.23732539081062502110840150965, 6.63906349572675573240747739258, 7.56241282060100871968041344083, 8.212973313819630460444369298086
