L(s) = 1 | + 1.06·3-s − 0.715·5-s − 1.86·9-s − 0.203·11-s − 4.07·13-s − 0.763·15-s + 1.49·17-s − 1.74·19-s − 4.09·23-s − 4.48·25-s − 5.18·27-s − 5.13·29-s − 3.89·31-s − 0.216·33-s + 9.19·37-s − 4.34·39-s − 41-s + 1.74·43-s + 1.33·45-s + 6.43·47-s + 1.59·51-s + 6.61·53-s + 0.145·55-s − 1.86·57-s + 12.8·59-s + 14.5·61-s + 2.91·65-s + ⋯ |
L(s) = 1 | + 0.615·3-s − 0.319·5-s − 0.620·9-s − 0.0612·11-s − 1.13·13-s − 0.197·15-s + 0.363·17-s − 0.400·19-s − 0.854·23-s − 0.897·25-s − 0.998·27-s − 0.953·29-s − 0.699·31-s − 0.0377·33-s + 1.51·37-s − 0.696·39-s − 0.156·41-s + 0.266·43-s + 0.198·45-s + 0.938·47-s + 0.223·51-s + 0.908·53-s + 0.0196·55-s − 0.246·57-s + 1.67·59-s + 1.85·61-s + 0.361·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.513232468\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.513232468\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 + 0.715T + 5T^{2} \) |
| 11 | \( 1 + 0.203T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 + 3.89T + 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 - 6.61T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 0.105T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 6.88T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.76757831034988734191946053654, −7.44103241580292112766046226329, −6.47931589183108057458256219595, −5.65282593299150694410314319720, −5.14370926187648510069233882154, −3.97126160142866225437000288699, −3.68697897424732854206218047969, −2.43319898561093060756990238707, −2.18603764386119010951183720362, −0.55481992855078839024431896340,
0.55481992855078839024431896340, 2.18603764386119010951183720362, 2.43319898561093060756990238707, 3.68697897424732854206218047969, 3.97126160142866225437000288699, 5.14370926187648510069233882154, 5.65282593299150694410314319720, 6.47931589183108057458256219595, 7.44103241580292112766046226329, 7.76757831034988734191946053654