L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 11-s − 12-s − 4·13-s − 16-s + 4·17-s + 18-s − 8·19-s − 22-s − 8·23-s − 3·24-s − 5·25-s − 4·26-s + 27-s + 2·29-s − 4·31-s + 5·32-s − 33-s + 4·34-s − 36-s + 10·37-s − 8·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.10·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.83·19-s − 0.213·22-s − 1.66·23-s − 0.612·24-s − 25-s − 0.784·26-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.174·33-s + 0.685·34-s − 1/6·36-s + 1.64·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.038956212657764719324110399957, −8.074730488157487416504173752135, −7.65723169668198833297114449990, −6.30975541870555345444458799385, −5.70041352880735109110133320738, −4.56293624923530839625501773863, −4.08923644740302608808427210611, −3.00132382947937649374672530191, −2.05220114561493272819870484669, 0,
2.05220114561493272819870484669, 3.00132382947937649374672530191, 4.08923644740302608808427210611, 4.56293624923530839625501773863, 5.70041352880735109110133320738, 6.30975541870555345444458799385, 7.65723169668198833297114449990, 8.074730488157487416504173752135, 9.038956212657764719324110399957