L(s) = 1 | + 2-s + 4-s + 5-s + 4·7-s + 8-s + 10-s + 2·11-s + 4·14-s + 16-s − 2·17-s + 20-s + 2·22-s − 23-s + 25-s + 4·28-s + 4·29-s + 32-s − 2·34-s + 4·35-s + 10·37-s + 40-s − 6·41-s + 2·43-s + 2·44-s − 46-s − 12·47-s + 9·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s + 0.426·22-s − 0.208·23-s + 1/5·25-s + 0.755·28-s + 0.742·29-s + 0.176·32-s − 0.342·34-s + 0.676·35-s + 1.64·37-s + 0.158·40-s − 0.937·41-s + 0.304·43-s + 0.301·44-s − 0.147·46-s − 1.75·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.739002811\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.739002811\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.076091713653181010715658475879, −8.208781540882969526831297540215, −7.60416359625391549446622435642, −6.54339823395422894886744855741, −5.96242520137679473951679895257, −4.79077329145906956219957278889, −4.59738734063357382886923854131, −3.33419874663372898406963550539, −2.16617393996501176803226728421, −1.34409171825676144708222230132,
1.34409171825676144708222230132, 2.16617393996501176803226728421, 3.33419874663372898406963550539, 4.59738734063357382886923854131, 4.79077329145906956219957278889, 5.96242520137679473951679895257, 6.54339823395422894886744855741, 7.60416359625391549446622435642, 8.208781540882969526831297540215, 9.076091713653181010715658475879