| L(s) = 1 | − 1.97·2-s + 2.89·4-s + 1.09·5-s − 3.73·8-s + 9-s − 2.15·10-s − 0.165·11-s + 0.490·13-s + 4.46·16-s + 1.57·17-s − 1.97·18-s + 3.16·20-s + 0.325·22-s − 1.35·23-s + 0.196·25-s − 0.968·26-s − 1.75·29-s − 5.08·32-s − 3.11·34-s + 2.89·36-s − 4.08·40-s − 0.477·44-s + 1.09·45-s + 2.67·46-s − 0.803·47-s + 49-s − 0.387·50-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.89·4-s + 1.09·5-s − 3.73·8-s + 9-s − 2.15·10-s − 0.165·11-s + 0.490·13-s + 4.46·16-s + 1.57·17-s − 1.97·18-s + 3.16·20-s + 0.325·22-s − 1.35·23-s + 0.196·25-s − 0.968·26-s − 1.75·29-s − 5.08·32-s − 3.11·34-s + 2.89·36-s − 4.08·40-s − 0.477·44-s + 1.09·45-s + 2.67·46-s − 0.803·47-s + 49-s − 0.387·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5728536173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5728536173\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 919 | \( 1 - T \) |
| good | 2 | \( 1 + 1.97T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.09T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.165T + T^{2} \) |
| 13 | \( 1 - 0.490T + T^{2} \) |
| 17 | \( 1 - 1.57T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.35T + T^{2} \) |
| 29 | \( 1 + 1.75T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.803T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.89T + T^{2} \) |
| 61 | \( 1 + 1.35T + T^{2} \) |
| 67 | \( 1 + 1.75T + T^{2} \) |
| 71 | \( 1 - 0.490T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.803T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.993300445940039750834384230690, −9.660822678636831691127753639381, −8.787203431747228783035206196046, −7.80034362278969066793663997568, −7.28557242101668910907976154969, −6.16924999531606456750044334754, −5.63633274141263291899387877502, −3.55140022204877144121022409514, −2.13982912183236701146907158787, −1.37203414209350851053724013141,
1.37203414209350851053724013141, 2.13982912183236701146907158787, 3.55140022204877144121022409514, 5.63633274141263291899387877502, 6.16924999531606456750044334754, 7.28557242101668910907976154969, 7.80034362278969066793663997568, 8.787203431747228783035206196046, 9.660822678636831691127753639381, 9.993300445940039750834384230690
