| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 3·11-s + 2·12-s − 6·13-s + 15-s + 4·16-s + 4·17-s + 19-s + 2·20-s − 21-s + 25-s − 27-s − 2·28-s − 2·29-s + 5·31-s − 3·33-s − 35-s − 2·36-s + 8·37-s + 6·39-s − 3·41-s − 2·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 1.66·13-s + 0.258·15-s + 16-s + 0.970·17-s + 0.229·19-s + 0.447·20-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.898·31-s − 0.522·33-s − 0.169·35-s − 1/3·36-s + 1.31·37-s + 0.960·39-s − 0.468·41-s − 0.304·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.72923252330281, −14.17373901537774, −13.72296428125252, −13.05889323675266, −12.47533979506291, −12.10304629120281, −11.76162433184114, −11.21636721911334, −10.40823584765177, −10.00336017165693, −9.522125936932963, −9.109531006035494, −8.340324377613346, −7.848607285200423, −7.397197567316381, −6.829391361009990, −6.010075600629322, −5.551761899853419, −4.766813792913939, −4.640791686900097, −3.921909465117057, −3.284213784487263, −2.501069357038603, −1.456104473204086, −0.8300816217756280, 0,
0.8300816217756280, 1.456104473204086, 2.501069357038603, 3.284213784487263, 3.921909465117057, 4.640791686900097, 4.766813792913939, 5.551761899853419, 6.010075600629322, 6.829391361009990, 7.397197567316381, 7.848607285200423, 8.340324377613346, 9.109531006035494, 9.522125936932963, 10.00336017165693, 10.40823584765177, 11.21636721911334, 11.76162433184114, 12.10304629120281, 12.47533979506291, 13.05889323675266, 13.72296428125252, 14.17373901537774, 14.72923252330281
