| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 2·11-s + 12-s − 4·13-s + 16-s + 17-s + 2·18-s − 6·19-s + 2·22-s − 23-s − 24-s + 4·26-s − 5·27-s + 29-s − 6·31-s − 32-s − 2·33-s − 34-s − 2·36-s − 2·37-s + 6·38-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.603·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 1.37·19-s + 0.426·22-s − 0.208·23-s − 0.204·24-s + 0.784·26-s − 0.962·27-s + 0.185·29-s − 1.07·31-s − 0.176·32-s − 0.348·33-s − 0.171·34-s − 1/3·36-s − 0.328·37-s + 0.973·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.29730425862645, −14.66381969850237, −14.47930865705244, −13.75565460489021, −13.12268306037941, −12.71221283456826, −12.02345609181312, −11.63949280064932, −10.90739293902742, −10.50697294174484, −9.915478311157257, −9.458181172507766, −8.830414416831144, −8.367720206001320, −7.933538588386642, −7.413835129399704, −6.729359387780266, −6.216504503597374, −5.410550298537845, −4.959919496329879, −4.129095543801316, −3.270528153382278, −2.841424069622199, −2.086230573173483, −1.612977670579252, 0, 0,
1.612977670579252, 2.086230573173483, 2.841424069622199, 3.270528153382278, 4.129095543801316, 4.959919496329879, 5.410550298537845, 6.216504503597374, 6.729359387780266, 7.413835129399704, 7.933538588386642, 8.367720206001320, 8.830414416831144, 9.458181172507766, 9.915478311157257, 10.50697294174484, 10.90739293902742, 11.63949280064932, 12.02345609181312, 12.71221283456826, 13.12268306037941, 13.75565460489021, 14.47930865705244, 14.66381969850237, 15.29730425862645
