| L(s) = 1 | + 2-s + 4-s − 3.53·5-s − 2.75·7-s + 8-s − 3.53·10-s + 2.94·11-s + 5.06·13-s − 2.75·14-s + 16-s − 4.36·19-s − 3.53·20-s + 2.94·22-s − 6·23-s + 7.47·25-s + 5.06·26-s − 2.75·28-s + 4.80·29-s + 0.716·31-s + 32-s + 9.74·35-s + 10.2·37-s − 4.36·38-s − 3.53·40-s − 12.5·41-s + 10.9·43-s + 2.94·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.57·5-s − 1.04·7-s + 0.353·8-s − 1.11·10-s + 0.887·11-s + 1.40·13-s − 0.737·14-s + 0.250·16-s − 1.00·19-s − 0.789·20-s + 0.627·22-s − 1.25·23-s + 1.49·25-s + 0.993·26-s − 0.521·28-s + 0.891·29-s + 0.128·31-s + 0.176·32-s + 1.64·35-s + 1.67·37-s − 0.708·38-s − 0.558·40-s − 1.96·41-s + 1.66·43-s + 0.443·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3.53T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 - 0.716T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 - 5.67T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 + 9.04T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.973152858192525349295937255721, −6.86047298075096528410456338082, −6.44594993882802597202029261880, −5.87891560142518876120498167612, −4.53939814674418161118729549074, −4.02888698182509702455385453652, −3.56647986835367920470993008290, −2.76351970463784161682687821235, −1.31725744105147816996898716784, 0,
1.31725744105147816996898716784, 2.76351970463784161682687821235, 3.56647986835367920470993008290, 4.02888698182509702455385453652, 4.53939814674418161118729549074, 5.87891560142518876120498167612, 6.44594993882802597202029261880, 6.86047298075096528410456338082, 7.973152858192525349295937255721
