| L(s) = 1 | + 2-s − 2.78·3-s + 4-s + 1.22·5-s − 2.78·6-s − 7-s + 8-s + 4.75·9-s + 1.22·10-s + 1.13·11-s − 2.78·12-s + 1.31·13-s − 14-s − 3.41·15-s + 16-s − 5.32·17-s + 4.75·18-s − 6.41·19-s + 1.22·20-s + 2.78·21-s + 1.13·22-s + 7.82·23-s − 2.78·24-s − 3.49·25-s + 1.31·26-s − 4.88·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.60·3-s + 0.5·4-s + 0.548·5-s − 1.13·6-s − 0.377·7-s + 0.353·8-s + 1.58·9-s + 0.388·10-s + 0.341·11-s − 0.803·12-s + 0.363·13-s − 0.267·14-s − 0.882·15-s + 0.250·16-s − 1.29·17-s + 1.12·18-s − 1.47·19-s + 0.274·20-s + 0.607·21-s + 0.241·22-s + 1.63·23-s − 0.568·24-s − 0.698·25-s + 0.257·26-s − 0.939·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 - 7.10T + 29T^{2} \) |
| 31 | \( 1 - 4.01T + 31T^{2} \) |
| 37 | \( 1 + 9.74T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 - 1.71T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 2.88T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 4.36T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 7.92T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.19578269566254138581744184853, −6.58338557322159206266577188054, −6.35941056468974066672709038097, −5.63891157344620115187418638841, −4.80147089073290699882626260064, −4.45246738669905847073484416235, −3.38372692422934359009953661256, −2.26810809488564176859041501207, −1.28838303786589172314287973223, 0,
1.28838303786589172314287973223, 2.26810809488564176859041501207, 3.38372692422934359009953661256, 4.45246738669905847073484416235, 4.80147089073290699882626260064, 5.63891157344620115187418638841, 6.35941056468974066672709038097, 6.58338557322159206266577188054, 7.19578269566254138581744184853
