| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 4·13-s − 14-s + 16-s − 4·19-s − 20-s + 3·23-s + 25-s + 4·26-s + 28-s + 6·29-s − 2·31-s − 32-s − 35-s + 4·37-s + 4·38-s + 40-s − 3·41-s − 4·43-s − 3·46-s + 12·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 0.625·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.169·35-s + 0.657·37-s + 0.648·38-s + 0.158·40-s − 0.468·41-s − 0.609·43-s − 0.442·46-s + 1.75·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078009054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078009054\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.31657803078497, −14.78789046254847, −14.55960695492344, −13.74007518210866, −13.07321963622318, −12.51407935012146, −11.96609360119241, −11.56425144815107, −10.89820612198467, −10.39781147345931, −9.926484146736894, −9.230954222816558, −8.669382652997079, −8.175339060216048, −7.662188395770253, −6.861718410471492, −6.748867949814229, −5.661548102147418, −5.122660260682672, −4.383120776705575, −3.796154538220933, −2.734864788020861, −2.370316096900203, −1.356182278145367, −0.4769141408428889,
0.4769141408428889, 1.356182278145367, 2.370316096900203, 2.734864788020861, 3.796154538220933, 4.383120776705575, 5.122660260682672, 5.661548102147418, 6.748867949814229, 6.861718410471492, 7.662188395770253, 8.175339060216048, 8.669382652997079, 9.230954222816558, 9.926484146736894, 10.39781147345931, 10.89820612198467, 11.56425144815107, 11.96609360119241, 12.51407935012146, 13.07321963622318, 13.74007518210866, 14.55960695492344, 14.78789046254847, 15.31657803078497
