| L(s) = 1 | − 0.567·2-s − 2.99·3-s − 1.67·4-s − 1.94·5-s + 1.69·6-s − 4.52·7-s + 2.08·8-s + 5.94·9-s + 1.10·10-s − 11-s + 5.01·12-s − 4.51·13-s + 2.56·14-s + 5.83·15-s + 2.17·16-s − 17-s − 3.37·18-s + 1.75·19-s + 3.27·20-s + 13.5·21-s + 0.567·22-s + 5.22·23-s − 6.24·24-s − 1.19·25-s + 2.55·26-s − 8.82·27-s + 7.59·28-s + ⋯ |
| L(s) = 1 | − 0.401·2-s − 1.72·3-s − 0.838·4-s − 0.871·5-s + 0.693·6-s − 1.71·7-s + 0.737·8-s + 1.98·9-s + 0.349·10-s − 0.301·11-s + 1.44·12-s − 1.25·13-s + 0.686·14-s + 1.50·15-s + 0.542·16-s − 0.242·17-s − 0.795·18-s + 0.403·19-s + 0.731·20-s + 2.95·21-s + 0.120·22-s + 1.09·23-s − 1.27·24-s − 0.239·25-s + 0.502·26-s − 1.69·27-s + 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.788045952\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.788045952\times10^{-5}\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.567T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 1.94T + 5T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 13 | \( 1 + 4.51T + 13T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 0.647T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 2.20T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 61 | \( 1 + 9.15T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 5.72T + 73T^{2} \) |
| 79 | \( 1 + 9.59T + 79T^{2} \) |
| 83 | \( 1 + 8.30T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 - 5.60T + 97T^{2} \) |
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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
−7.53386775180075125494445104412, −7.18200314393853346441956490034, −6.44766870703666586261881785861, −5.76850721858532673213368315232, −4.95699200635405583394171269645, −4.54784884453139442571349272051, −3.68387969333176487421791336611, −2.82534180521311531641515426637, −1.11647139947833132256515396026, −0.00429294570874279149769589547,
0.00429294570874279149769589547, 1.11647139947833132256515396026, 2.82534180521311531641515426637, 3.68387969333176487421791336611, 4.54784884453139442571349272051, 4.95699200635405583394171269645, 5.76850721858532673213368315232, 6.44766870703666586261881785861, 7.18200314393853346441956490034, 7.53386775180075125494445104412
