| L(s) = 1 | − 3-s − 2·4-s − 4·5-s − 7-s + 9-s + 11-s + 2·12-s + 13-s + 4·15-s + 4·16-s − 17-s + 8·20-s + 21-s + 8·23-s + 11·25-s − 27-s + 2·28-s − 2·29-s + 2·31-s − 33-s + 4·35-s − 2·36-s + 4·37-s − 39-s + 6·43-s − 2·44-s − 4·45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.277·13-s + 1.03·15-s + 16-s − 0.242·17-s + 1.78·20-s + 0.218·21-s + 1.66·23-s + 11/5·25-s − 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.359·31-s − 0.174·33-s + 0.676·35-s − 1/3·36-s + 0.657·37-s − 0.160·39-s + 0.914·43-s − 0.301·44-s − 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 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
−14.34469840451133, −13.51954542738980, −13.07212391177653, −12.72957368582720, −12.22081196906687, −11.71562110132153, −11.28405370507214, −10.81612748805886, −10.34542374301255, −9.586909399289648, −9.107978014744912, −8.656169103920996, −8.188100813481979, −7.522398006217228, −7.192187648087592, −6.574768234653681, −5.920897970773408, −5.200061078978076, −4.758860958527438, −4.178444524010499, −3.804550494655279, −3.267823389587707, −2.544301239394435, −1.145226911476732, −0.7549445779419052, 0,
0.7549445779419052, 1.145226911476732, 2.544301239394435, 3.267823389587707, 3.804550494655279, 4.178444524010499, 4.758860958527438, 5.200061078978076, 5.920897970773408, 6.574768234653681, 7.192187648087592, 7.522398006217228, 8.188100813481979, 8.656169103920996, 9.107978014744912, 9.586909399289648, 10.34542374301255, 10.81612748805886, 11.28405370507214, 11.71562110132153, 12.22081196906687, 12.72957368582720, 13.07212391177653, 13.51954542738980, 14.34469840451133
