| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3·7-s + 8-s + 9-s − 3·11-s − 12-s − 3·13-s − 3·14-s + 16-s − 2·17-s + 18-s − 5·19-s + 3·21-s − 3·22-s − 24-s − 3·26-s − 27-s − 3·28-s + 9·29-s + 2·31-s + 32-s + 3·33-s − 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.14·19-s + 0.654·21-s − 0.639·22-s − 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.566·28-s + 1.67·29-s + 0.359·31-s + 0.176·32-s + 0.522·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.49062551454206, −13.80936429780053, −13.39246014448166, −13.05846617709016, −12.51558284454747, −12.03161551256460, −11.83782088785196, −10.95864698138916, −10.46496188330460, −10.12630530959341, −9.819274672307636, −8.821581947798909, −8.499095748373296, −7.759598868200512, −7.087893502020193, −6.564727525289281, −6.447739588794306, −5.636046900310310, −5.044940393457936, −4.704073071203879, −4.046650027745624, −3.277203923035918, −2.811105201005630, −2.201249817580216, −1.382950665585472, 0, 0,
1.382950665585472, 2.201249817580216, 2.811105201005630, 3.277203923035918, 4.046650027745624, 4.704073071203879, 5.044940393457936, 5.636046900310310, 6.447739588794306, 6.564727525289281, 7.087893502020193, 7.759598868200512, 8.499095748373296, 8.821581947798909, 9.819274672307636, 10.12630530959341, 10.46496188330460, 10.95864698138916, 11.83782088785196, 12.03161551256460, 12.51558284454747, 13.05846617709016, 13.39246014448166, 13.80936429780053, 14.49062551454206
