| L(s) = 1 | − 3-s − 2·4-s + 4·5-s − 4·7-s + 9-s + 4·11-s + 2·12-s − 5·13-s − 4·15-s + 4·16-s − 5·17-s + 5·19-s − 8·20-s + 4·21-s + 23-s + 11·25-s − 27-s + 8·28-s − 29-s − 2·31-s − 4·33-s − 16·35-s − 2·36-s + 5·37-s + 5·39-s − 2·41-s + 43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.78·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 1.38·13-s − 1.03·15-s + 16-s − 1.21·17-s + 1.14·19-s − 1.78·20-s + 0.872·21-s + 0.208·23-s + 11/5·25-s − 0.192·27-s + 1.51·28-s − 0.185·29-s − 0.359·31-s − 0.696·33-s − 2.70·35-s − 1/3·36-s + 0.821·37-s + 0.800·39-s − 0.312·41-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223221246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223221246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−9.260867415265607682796848286962, −8.964560335588406953513666380604, −7.28994226615198199606122345411, −6.60123660484370742188522259092, −5.96397978578636854059591521654, −5.27029940034070622731192453076, −4.43152301254281187157853772822, −3.28897286527146816079042547184, −2.16848873486685433534249006869, −0.76225775874813821996698126597,
0.76225775874813821996698126597, 2.16848873486685433534249006869, 3.28897286527146816079042547184, 4.43152301254281187157853772822, 5.27029940034070622731192453076, 5.96397978578636854059591521654, 6.60123660484370742188522259092, 7.28994226615198199606122345411, 8.964560335588406953513666380604, 9.260867415265607682796848286962
