| L(s) = 1 | − 2-s − 1.39·3-s + 4-s + 4.04·5-s + 1.39·6-s − 1.52·7-s − 8-s − 1.06·9-s − 4.04·10-s + 1.03·11-s − 1.39·12-s − 2.91·13-s + 1.52·14-s − 5.62·15-s + 16-s + 1.25·17-s + 1.06·18-s − 19-s + 4.04·20-s + 2.12·21-s − 1.03·22-s − 7.44·23-s + 1.39·24-s + 11.3·25-s + 2.91·26-s + 5.65·27-s − 1.52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.802·3-s + 0.5·4-s + 1.80·5-s + 0.567·6-s − 0.577·7-s − 0.353·8-s − 0.355·9-s − 1.27·10-s + 0.313·11-s − 0.401·12-s − 0.809·13-s + 0.408·14-s − 1.45·15-s + 0.250·16-s + 0.303·17-s + 0.251·18-s − 0.229·19-s + 0.904·20-s + 0.463·21-s − 0.221·22-s − 1.55·23-s + 0.283·24-s + 2.27·25-s + 0.572·26-s + 1.08·27-s − 0.288·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.121302727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.121302727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 - 6.98T + 47T^{2} \) |
| 53 | \( 1 + 2.00T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 3.70T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 + 9.09T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.901423879387862091081411145713, −6.83046503140593058580307637811, −6.42064293827861438833920495357, −5.91258686842110132182650235648, −5.35348057687181216583209408992, −4.54492024121746812134458556129, −3.18817175251146515510583754516, −2.43331899057364962924183751738, −1.70935210796601067086154613232, −0.59944948839891428879207098540,
0.59944948839891428879207098540, 1.70935210796601067086154613232, 2.43331899057364962924183751738, 3.18817175251146515510583754516, 4.54492024121746812134458556129, 5.35348057687181216583209408992, 5.91258686842110132182650235648, 6.42064293827861438833920495357, 6.83046503140593058580307637811, 7.901423879387862091081411145713
