| L(s) = 1 | + 2·5-s + 11-s + 2·13-s + 4·17-s − 6·19-s − 25-s + 8·29-s − 8·31-s + 10·37-s + 8·41-s + 2·43-s + 8·47-s + 2·53-s + 2·55-s − 12·59-s − 10·61-s + 4·65-s − 12·67-s + 8·71-s − 6·73-s + 2·79-s − 16·83-s + 8·85-s − 14·89-s − 12·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s − 1/5·25-s + 1.48·29-s − 1.43·31-s + 1.64·37-s + 1.24·41-s + 0.304·43-s + 1.16·47-s + 0.274·53-s + 0.269·55-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 1.46·67-s + 0.949·71-s − 0.702·73-s + 0.225·79-s − 1.75·83-s + 0.867·85-s − 1.48·89-s − 1.23·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.18988415139971, −13.85092646650329, −13.32111313994153, −12.74115248599868, −12.40705039329489, −11.90124165750013, −11.05382491975315, −10.84030094732727, −10.28162226887667, −9.717446589462990, −9.254198319333990, −8.821813429956403, −8.201130206568439, −7.649685480556141, −7.112568650479981, −6.333268449881420, −5.937223888702586, −5.731240679515176, −4.797456227540916, −4.238388656081238, −3.768328792255835, −2.796233625021225, −2.488962314310414, −1.529011894761435, −1.145536077234290, 0,
1.145536077234290, 1.529011894761435, 2.488962314310414, 2.796233625021225, 3.768328792255835, 4.238388656081238, 4.797456227540916, 5.731240679515176, 5.937223888702586, 6.333268449881420, 7.112568650479981, 7.649685480556141, 8.201130206568439, 8.821813429956403, 9.254198319333990, 9.717446589462990, 10.28162226887667, 10.84030094732727, 11.05382491975315, 11.90124165750013, 12.40705039329489, 12.74115248599868, 13.32111313994153, 13.85092646650329, 14.18988415139971
