| L(s) = 1 | − 1.67·2-s + 3.39·3-s + 0.790·4-s − 1.45·5-s − 5.67·6-s − 1.59·7-s + 2.02·8-s + 8.55·9-s + 2.43·10-s + 11-s + 2.68·12-s − 1.30·13-s + 2.65·14-s − 4.95·15-s − 4.95·16-s − 2.95·17-s − 14.2·18-s − 3.43·19-s − 1.15·20-s − 5.40·21-s − 1.67·22-s + 6.86·24-s − 2.87·25-s + 2.18·26-s + 18.8·27-s − 1.25·28-s + 6.51·29-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 1.96·3-s + 0.395·4-s − 0.652·5-s − 2.31·6-s − 0.601·7-s + 0.714·8-s + 2.85·9-s + 0.770·10-s + 0.301·11-s + 0.775·12-s − 0.362·13-s + 0.710·14-s − 1.28·15-s − 1.23·16-s − 0.715·17-s − 3.36·18-s − 0.787·19-s − 0.257·20-s − 1.18·21-s − 0.356·22-s + 1.40·24-s − 0.574·25-s + 0.428·26-s + 3.63·27-s − 0.237·28-s + 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682043502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682043502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 - 3.39T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 2.95T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 29 | \( 1 - 6.51T + 29T^{2} \) |
| 31 | \( 1 - 1.28T + 31T^{2} \) |
| 37 | \( 1 + 0.915T + 37T^{2} \) |
| 41 | \( 1 - 0.847T + 41T^{2} \) |
| 43 | \( 1 + 4.84T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 9.11T + 53T^{2} \) |
| 59 | \( 1 + 7.33T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 0.274T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.202885115424736677154876113534, −7.87353951952110271613521010452, −6.83677796041936665527704159668, −6.74210915620981813508299439015, −4.88164363527641782497933429397, −4.10363634904805869727048527668, −3.62455301171581007444635412912, −2.54801720185589122508596597775, −1.97308774231776513143266408941, −0.74386488038721663259618992562,
0.74386488038721663259618992562, 1.97308774231776513143266408941, 2.54801720185589122508596597775, 3.62455301171581007444635412912, 4.10363634904805869727048527668, 4.88164363527641782497933429397, 6.74210915620981813508299439015, 6.83677796041936665527704159668, 7.87353951952110271613521010452, 8.202885115424736677154876113534
