| L(s) = 1 | + 1.94·2-s − 3-s + 1.79·4-s − 1.40·5-s − 1.94·6-s − 0.394·8-s + 9-s − 2.73·10-s + 4.53·11-s − 1.79·12-s − 2.35·13-s + 1.40·15-s − 4.36·16-s − 1.10·17-s + 1.94·18-s + 0.638·19-s − 2.51·20-s + 8.83·22-s − 23-s + 0.394·24-s − 3.03·25-s − 4.58·26-s − 27-s + 2.63·29-s + 2.73·30-s − 5.19·31-s − 7.71·32-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 0.577·3-s + 0.898·4-s − 0.626·5-s − 0.795·6-s − 0.139·8-s + 0.333·9-s − 0.863·10-s + 1.36·11-s − 0.518·12-s − 0.653·13-s + 0.361·15-s − 1.09·16-s − 0.268·17-s + 0.459·18-s + 0.146·19-s − 0.563·20-s + 1.88·22-s − 0.208·23-s + 0.0805·24-s − 0.607·25-s − 0.900·26-s − 0.192·27-s + 0.488·29-s + 0.498·30-s − 0.932·31-s − 1.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 - 0.638T + 19T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 + 0.821T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 1.99T + 73T^{2} \) |
| 79 | \( 1 - 4.18T + 79T^{2} \) |
| 83 | \( 1 - 8.69T + 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
| 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
−8.029046902383387840310852480912, −7.19541702693760076618317717328, −6.48648815738169165954002361437, −5.93521748222247805421140709523, −5.02150356951414563242717263094, −4.35847862652766712527195298254, −3.81572321686217155561849092033, −2.92020139751084833661808842582, −1.63416442384464095490505608930, 0,
1.63416442384464095490505608930, 2.92020139751084833661808842582, 3.81572321686217155561849092033, 4.35847862652766712527195298254, 5.02150356951414563242717263094, 5.93521748222247805421140709523, 6.48648815738169165954002361437, 7.19541702693760076618317717328, 8.029046902383387840310852480912
