L(s) = 1 | − 2.60·2-s + 3.31·3-s + 4.76·4-s − 1.17·5-s − 8.61·6-s + 2.01·7-s − 7.19·8-s + 7.97·9-s + 3.04·10-s − 2.35·11-s + 15.7·12-s − 1.55·13-s − 5.24·14-s − 3.88·15-s + 9.17·16-s − 5.14·17-s − 20.7·18-s + 1.38·19-s − 5.58·20-s + 6.67·21-s + 6.13·22-s − 7.77·23-s − 23.8·24-s − 3.62·25-s + 4.04·26-s + 16.5·27-s + 9.60·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 1.91·3-s + 2.38·4-s − 0.523·5-s − 3.51·6-s + 0.761·7-s − 2.54·8-s + 2.65·9-s + 0.963·10-s − 0.710·11-s + 4.55·12-s − 0.431·13-s − 1.40·14-s − 1.00·15-s + 2.29·16-s − 1.24·17-s − 4.89·18-s + 0.317·19-s − 1.24·20-s + 1.45·21-s + 1.30·22-s − 1.62·23-s − 4.86·24-s − 0.725·25-s + 0.792·26-s + 3.17·27-s + 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 - 0.340T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 - 0.204T + 97T^{2} \) |
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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
−8.140491200277348474738122400619, −7.86233162047528974564827462553, −7.20527065284169204439278138412, −6.45914657428413412444443240099, −4.90789808277856015784990441864, −3.90073449826927615774067715547, −2.99593205778721294313606581874, −2.04529931917132111712382546938, −1.74799055274528024228995111296, 0,
1.74799055274528024228995111296, 2.04529931917132111712382546938, 2.99593205778721294313606581874, 3.90073449826927615774067715547, 4.90789808277856015784990441864, 6.45914657428413412444443240099, 7.20527065284169204439278138412, 7.86233162047528974564827462553, 8.140491200277348474738122400619