L(s) = 1 | − 2·2-s + 2·4-s − 3·5-s − 4·7-s + 6·10-s − 6·11-s − 7·13-s + 8·14-s − 4·16-s + 2·17-s − 6·20-s + 12·22-s − 23-s + 4·25-s + 14·26-s − 8·28-s − 9·29-s − 5·31-s + 8·32-s − 4·34-s + 12·35-s − 3·37-s + 6·41-s + 4·43-s − 12·44-s + 2·46-s − 2·47-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 1.34·5-s − 1.51·7-s + 1.89·10-s − 1.80·11-s − 1.94·13-s + 2.13·14-s − 16-s + 0.485·17-s − 1.34·20-s + 2.55·22-s − 0.208·23-s + 4/5·25-s + 2.74·26-s − 1.51·28-s − 1.67·29-s − 0.898·31-s + 1.41·32-s − 0.685·34-s + 2.02·35-s − 0.493·37-s + 0.937·41-s + 0.609·43-s − 1.80·44-s + 0.294·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.173685595792635956288106677373, −7.979069184297479970157387427743, −7.50237058980749934967978419558, −7.14586714404397132780454647934, −5.67487786880756741852024245547, −4.56876609912340916029501181301, −3.31695645330805523753956502481, −2.38337836408984736978763024018, 0, 0,
2.38337836408984736978763024018, 3.31695645330805523753956502481, 4.56876609912340916029501181301, 5.67487786880756741852024245547, 7.14586714404397132780454647934, 7.50237058980749934967978419558, 7.979069184297479970157387427743, 9.173685595792635956288106677373