| L(s) = 1 | − 0.512·2-s − 1.73·4-s + 0.842·5-s − 1.17·7-s + 1.91·8-s − 0.432·10-s + 5.49·11-s − 1.04·13-s + 0.601·14-s + 2.49·16-s + 6.90·17-s + 19-s − 1.46·20-s − 2.81·22-s − 3.42·23-s − 4.29·25-s + 0.534·26-s + 2.03·28-s + 4.94·29-s + 9.24·31-s − 5.11·32-s − 3.54·34-s − 0.987·35-s + 8.60·37-s − 0.512·38-s + 1.61·40-s + 1.68·41-s + ⋯ |
| L(s) = 1 | − 0.362·2-s − 0.868·4-s + 0.376·5-s − 0.442·7-s + 0.677·8-s − 0.136·10-s + 1.65·11-s − 0.289·13-s + 0.160·14-s + 0.622·16-s + 1.67·17-s + 0.229·19-s − 0.327·20-s − 0.600·22-s − 0.713·23-s − 0.858·25-s + 0.104·26-s + 0.384·28-s + 0.918·29-s + 1.66·31-s − 0.903·32-s − 0.607·34-s − 0.166·35-s + 1.41·37-s − 0.0832·38-s + 0.255·40-s + 0.263·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737286521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737286521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.512T + 2T^{2} \) |
| 5 | \( 1 - 0.842T + 5T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 7.64T + 43T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 6.62T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 0.554T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 + 7.25T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.947549810088308380514594258623, −7.28656242528208853369635517299, −6.20826760144123785608311634006, −6.00632284561203019502223064597, −4.96502362066793257563873639664, −4.22064522185033102354352354802, −3.64615230662474924266959597347, −2.71262251830807264139556482892, −1.42219065185236129557312542814, −0.789402614023222416087761054345,
0.789402614023222416087761054345, 1.42219065185236129557312542814, 2.71262251830807264139556482892, 3.64615230662474924266959597347, 4.22064522185033102354352354802, 4.96502362066793257563873639664, 6.00632284561203019502223064597, 6.20826760144123785608311634006, 7.28656242528208853369635517299, 7.947549810088308380514594258623
