| L(s) = 1 | − 0.209·2-s − 3-s − 1.95·4-s + 0.209·6-s + 0.591·7-s + 0.827·8-s + 9-s − 0.870·11-s + 1.95·12-s + 1.15·13-s − 0.123·14-s + 3.73·16-s − 4.93·17-s − 0.209·18-s − 2.84·19-s − 0.591·21-s + 0.182·22-s + 6.91·23-s − 0.827·24-s − 0.240·26-s − 27-s − 1.15·28-s − 7.48·29-s + 3.45·31-s − 2.43·32-s + 0.870·33-s + 1.03·34-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 0.577·3-s − 0.978·4-s + 0.0853·6-s + 0.223·7-s + 0.292·8-s + 0.333·9-s − 0.262·11-s + 0.564·12-s + 0.319·13-s − 0.0330·14-s + 0.934·16-s − 1.19·17-s − 0.0492·18-s − 0.653·19-s − 0.128·21-s + 0.0388·22-s + 1.44·23-s − 0.168·24-s − 0.0471·26-s − 0.192·27-s − 0.218·28-s − 1.39·29-s + 0.621·31-s − 0.430·32-s + 0.151·33-s + 0.176·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.209T + 2T^{2} \) |
| 7 | \( 1 - 0.591T + 7T^{2} \) |
| 11 | \( 1 + 0.870T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 + 2.81T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 7.60T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 2.98T + 73T^{2} \) |
| 79 | \( 1 - 3.33T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 - 0.645T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
| 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
−9.104420090777818617354725582031, −8.076325453989171989904704369344, −7.36852667471047721159185492280, −6.31895140748792489154506104026, −5.59059088843827877611302350792, −4.59969188440715052594304183628, −4.19831349910687715367917400501, −2.81598929201090046958090229333, −1.34044347014290301584258655770, 0,
1.34044347014290301584258655770, 2.81598929201090046958090229333, 4.19831349910687715367917400501, 4.59969188440715052594304183628, 5.59059088843827877611302350792, 6.31895140748792489154506104026, 7.36852667471047721159185492280, 8.076325453989171989904704369344, 9.104420090777818617354725582031
