| L(s) = 1 | + 2.40·2-s − 2.55·3-s + 3.77·4-s − 6.13·6-s − 1.03·7-s + 4.26·8-s + 3.51·9-s + 4.74·11-s − 9.63·12-s + 5.35·13-s − 2.48·14-s + 2.69·16-s − 1.99·17-s + 8.44·18-s + 3.89·19-s + 2.64·21-s + 11.4·22-s − 1.23·23-s − 10.8·24-s + 12.8·26-s − 1.31·27-s − 3.90·28-s + 10.1·29-s − 31-s − 2.05·32-s − 12.1·33-s − 4.80·34-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 1.47·3-s + 1.88·4-s − 2.50·6-s − 0.391·7-s + 1.50·8-s + 1.17·9-s + 1.43·11-s − 2.78·12-s + 1.48·13-s − 0.664·14-s + 0.673·16-s − 0.484·17-s + 1.99·18-s + 0.893·19-s + 0.576·21-s + 2.43·22-s − 0.258·23-s − 2.22·24-s + 2.52·26-s − 0.253·27-s − 0.738·28-s + 1.88·29-s − 0.179·31-s − 0.362·32-s − 2.10·33-s − 0.823·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.742370990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.742370990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 + 2.55T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 + 1.99T + 17T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 + 2.69T + 41T^{2} \) |
| 43 | \( 1 + 7.63T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 - 4.80T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + 2.13T + 89T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
| 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
−10.85689525250034745905408624703, −9.796159437028752335297388268544, −8.551066027820885440265325752532, −6.91037125924436386783793937585, −6.39683195067869645544047945207, −5.92932238122486256930217365203, −4.91751545694964722683671090148, −4.12391299144726020213336617933, −3.20040869304088837406817234650, −1.30638929328000054966871809971,
1.30638929328000054966871809971, 3.20040869304088837406817234650, 4.12391299144726020213336617933, 4.91751545694964722683671090148, 5.92932238122486256930217365203, 6.39683195067869645544047945207, 6.91037125924436386783793937585, 8.551066027820885440265325752532, 9.796159437028752335297388268544, 10.85689525250034745905408624703
