L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s + 9-s + 11-s − 2·12-s − 15-s + 4·16-s + 2·17-s − 3·19-s + 2·20-s − 21-s + 23-s + 25-s + 27-s + 2·28-s − 6·29-s + 33-s + 35-s − 2·36-s + 6·37-s − 3·41-s − 6·43-s − 2·44-s − 45-s − 7·47-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.258·15-s + 16-s + 0.485·17-s − 0.688·19-s + 0.447·20-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.11·29-s + 0.174·33-s + 0.169·35-s − 1/3·36-s + 0.986·37-s − 0.468·41-s − 0.914·43-s − 0.301·44-s − 0.149·45-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.506387128275588003143113836870, −8.041039730129038287857669382661, −7.18499891999489484849614621021, −6.25933295009046497043879379845, −5.28672403925960649649380684511, −4.39232418345082500019570680695, −3.72102733264847901844190355391, −2.93897094458045874000317933199, −1.47553802685880125083223156910, 0,
1.47553802685880125083223156910, 2.93897094458045874000317933199, 3.72102733264847901844190355391, 4.39232418345082500019570680695, 5.28672403925960649649380684511, 6.25933295009046497043879379845, 7.18499891999489484849614621021, 8.041039730129038287857669382661, 8.506387128275588003143113836870