L(s) = 1 | − 1.95·2-s − 3-s + 1.82·4-s + 1.95·6-s + 4.57·7-s + 0.338·8-s + 9-s − 4.16·11-s − 1.82·12-s + 3.75·13-s − 8.94·14-s − 4.31·16-s − 3.26·17-s − 1.95·18-s − 0.243·19-s − 4.57·21-s + 8.14·22-s − 0.654·23-s − 0.338·24-s − 7.34·26-s − 27-s + 8.35·28-s − 8.54·29-s − 3.90·31-s + 7.76·32-s + 4.16·33-s + 6.39·34-s + ⋯ |
L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.913·4-s + 0.798·6-s + 1.72·7-s + 0.119·8-s + 0.333·9-s − 1.25·11-s − 0.527·12-s + 1.04·13-s − 2.39·14-s − 1.07·16-s − 0.792·17-s − 0.461·18-s − 0.0558·19-s − 0.998·21-s + 1.73·22-s − 0.136·23-s − 0.0690·24-s − 1.44·26-s − 0.192·27-s + 1.57·28-s − 1.58·29-s − 0.701·31-s + 1.37·32-s + 0.725·33-s + 1.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 4.16T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + 0.243T + 19T^{2} \) |
| 23 | \( 1 + 0.654T + 23T^{2} \) |
| 29 | \( 1 + 8.54T + 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.769T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 0.316T + 67T^{2} \) |
| 71 | \( 1 - 0.446T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 - 8.86T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 8.15T + 97T^{2} \) |
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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
−8.679545916761137844311466291091, −8.181218389720093471947511833719, −7.55603301121187014696962838829, −6.78095581366020259871362525636, −5.51410301829354224537251271185, −4.97322508897565073991685188899, −3.92367179937230769354795801908, −2.13562660544633195124668524043, −1.44806255035389705206448920758, 0,
1.44806255035389705206448920758, 2.13562660544633195124668524043, 3.92367179937230769354795801908, 4.97322508897565073991685188899, 5.51410301829354224537251271185, 6.78095581366020259871362525636, 7.55603301121187014696962838829, 8.181218389720093471947511833719, 8.679545916761137844311466291091