L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s − 2·9-s − 2·10-s + 11-s + 2·12-s + 4·13-s − 4·14-s − 15-s − 4·16-s − 2·17-s − 4·18-s − 2·20-s − 2·21-s + 2·22-s + 23-s − 4·25-s + 8·26-s − 5·27-s − 4·28-s − 2·30-s − 7·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s + 1.10·13-s − 1.06·14-s − 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s − 0.447·20-s − 0.436·21-s + 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.962·27-s − 0.755·28-s − 0.365·30-s − 1.25·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30899 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30899 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.486786116\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.486786116\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 53 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−15.03683152891530, −14.41939793496089, −14.06371471895509, −13.44713027512324, −13.19220463817368, −12.61666971372406, −12.07676976636816, −11.37889900465128, −11.20264311253609, −10.47187846080322, −9.509185355129794, −9.097041945583492, −8.695157772416545, −7.897483464878132, −7.360957520789839, −6.497372453388345, −6.158943406085272, −5.635915127517986, −4.914197217563140, −4.088473878158438, −3.683810004645797, −3.291688991697498, −2.562736860041089, −1.869542393771606, −0.5154294398057512,
0.5154294398057512, 1.869542393771606, 2.562736860041089, 3.291688991697498, 3.683810004645797, 4.088473878158438, 4.914197217563140, 5.635915127517986, 6.158943406085272, 6.497372453388345, 7.360957520789839, 7.897483464878132, 8.695157772416545, 9.097041945583492, 9.509185355129794, 10.47187846080322, 11.20264311253609, 11.37889900465128, 12.07676976636816, 12.61666971372406, 13.19220463817368, 13.44713027512324, 14.06371471895509, 14.41939793496089, 15.03683152891530