L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 6·11-s + 12-s − 4·13-s + 16-s − 3·17-s − 18-s + 4·19-s − 6·22-s − 3·23-s − 24-s + 4·26-s + 27-s − 6·29-s − 5·31-s − 32-s + 6·33-s + 3·34-s + 36-s − 8·37-s − 4·38-s − 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.917·19-s − 1.27·22-s − 0.625·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.176·32-s + 1.04·33-s + 0.514·34-s + 1/6·36-s − 1.31·37-s − 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.45773119963171127189424052540, −7.13157350532036700624325243994, −6.40182195849577572961054770108, −5.56800805358585562913075039277, −4.58401036071441833670910790812, −3.80454654589416254588003816875, −3.09542637144155554586290651848, −2.01150208958649418424553581434, −1.44253173014801006992293197684, 0,
1.44253173014801006992293197684, 2.01150208958649418424553581434, 3.09542637144155554586290651848, 3.80454654589416254588003816875, 4.58401036071441833670910790812, 5.56800805358585562913075039277, 6.40182195849577572961054770108, 7.13157350532036700624325243994, 7.45773119963171127189424052540