| L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.142 + 0.438i)3-s + (0.309 + 0.951i)4-s + (−2.07 + 0.843i)5-s + (0.142 − 0.438i)6-s + 3.64·7-s + (0.309 − 0.951i)8-s + (2.25 − 1.63i)9-s + (2.17 + 0.535i)10-s + (−4.35 − 3.16i)11-s + (−0.373 + 0.271i)12-s + (−1.25 + 0.914i)13-s + (−2.94 − 2.14i)14-s + (−0.665 − 0.788i)15-s + (−0.809 + 0.587i)16-s + (1.21 − 3.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.0823 + 0.253i)3-s + (0.154 + 0.475i)4-s + (−0.926 + 0.377i)5-s + (0.0582 − 0.179i)6-s + 1.37·7-s + (0.109 − 0.336i)8-s + (0.751 − 0.546i)9-s + (0.686 + 0.169i)10-s + (−1.31 − 0.954i)11-s + (−0.107 + 0.0783i)12-s + (−0.349 + 0.253i)13-s + (−0.787 − 0.572i)14-s + (−0.171 − 0.203i)15-s + (−0.202 + 0.146i)16-s + (0.295 − 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.842401 - 0.599613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.842401 - 0.599613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (2.07 - 0.843i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| good | 3 | \( 1 + (-0.142 - 0.438i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + (4.35 + 3.16i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.25 - 0.914i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.21 + 3.75i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (3.02 + 2.20i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.73 - 5.33i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.01 + 3.13i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.55 - 1.12i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.84 + 5.70i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + (2.68 + 8.24i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.34 + 10.2i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.86 + 3.53i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (10.1 + 7.40i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.02 + 6.24i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.85 - 5.72i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.59 + 1.88i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.67 - 14.3i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.437 - 1.34i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-13.0 - 9.49i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.702 - 2.16i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.02458693446495341661913192492, −8.986721704119516847976377292393, −8.089146426235113074720436071183, −7.71273425498337333362340544959, −6.78806620015641418335229492947, −5.29473447660936504366999519620, −4.41075357174859038540970654936, −3.42303044328258165705851705540, −2.30058439697406406239195937118, −0.64931534871098794592271257397,
1.30405549112262629463010064986, 2.43494007790560906841375544969, 4.40350482055061282611970085496, 4.77198573293292328929533238191, 5.89836313060119034395772246916, 7.48671182459732553405084844164, 7.66682003859289104548210911633, 8.121071434705465588626894010450, 9.219939237404288549773005156906, 10.37632422248274657879500090849
