L(s) = 1 | + 2-s − 4-s + 5-s + 7-s − 3·8-s − 3·9-s + 10-s + 6·13-s + 14-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 20-s − 8·23-s + 25-s + 6·26-s − 28-s + 10·29-s − 4·31-s + 5·32-s − 6·34-s + 35-s + 3·36-s + 6·37-s + 4·38-s − 3·40-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.377·7-s − 1.06·8-s − 9-s + 0.316·10-s + 1.66·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.85·29-s − 0.718·31-s + 0.883·32-s − 1.02·34-s + 0.169·35-s + 1/2·36-s + 0.986·37-s + 0.648·38-s − 0.474·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.288050517\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.288050517\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.564624605217639120859682087576, −7.85880887536054419534676088661, −6.51568395178496581172890040306, −6.06661044283277712738867145325, −5.49291750801948910467616648850, −4.59752674990842071518091776003, −3.93693373996263352320985772711, −3.07677740393696618525124124761, −2.15976114084922275451922909685, −0.77175110263009901472834259479,
0.77175110263009901472834259479, 2.15976114084922275451922909685, 3.07677740393696618525124124761, 3.93693373996263352320985772711, 4.59752674990842071518091776003, 5.49291750801948910467616648850, 6.06661044283277712738867145325, 6.51568395178496581172890040306, 7.85880887536054419534676088661, 8.564624605217639120859682087576