| L(s) = 1 | − 3-s − 5-s − 5·7-s + 9-s − 13-s + 15-s + 3·17-s − 4·19-s + 5·21-s − 8·23-s + 25-s − 27-s + 29-s − 2·31-s + 5·35-s − 8·37-s + 39-s − 6·41-s − 3·43-s − 45-s + 8·47-s + 18·49-s − 3·51-s − 12·53-s + 4·57-s + 8·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.727·17-s − 0.917·19-s + 1.09·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s − 0.359·31-s + 0.845·35-s − 1.31·37-s + 0.160·39-s − 0.937·41-s − 0.457·43-s − 0.149·45-s + 1.16·47-s + 18/7·49-s − 0.420·51-s − 1.64·53-s + 0.529·57-s + 1.04·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.23537498086210, −12.61489778058421, −12.50565924470414, −12.09242146426623, −11.56785928938383, −10.99927469150299, −10.36803810820297, −9.981581862093863, −9.875076063185350, −9.142326317845594, −8.570371400658234, −8.163193542298316, −7.426562416016153, −7.008883394619942, −6.544782274474040, −6.177232304207014, −5.560813650017252, −5.221279307715759, −4.250986440216502, −3.969870864386905, −3.418622607653490, −2.889382426127686, −2.190183026766750, −1.462809112708377, −0.4535304053493525, 0,
0.4535304053493525, 1.462809112708377, 2.190183026766750, 2.889382426127686, 3.418622607653490, 3.969870864386905, 4.250986440216502, 5.221279307715759, 5.560813650017252, 6.177232304207014, 6.544782274474040, 7.008883394619942, 7.426562416016153, 8.163193542298316, 8.570371400658234, 9.142326317845594, 9.875076063185350, 9.981581862093863, 10.36803810820297, 10.99927469150299, 11.56785928938383, 12.09242146426623, 12.50565924470414, 12.61489778058421, 13.23537498086210
