| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 7-s + 8-s − 2·9-s − 3·10-s − 12-s + 13-s + 14-s + 3·15-s + 16-s + 2·17-s − 2·18-s + 5·19-s − 3·20-s − 21-s + 23-s − 24-s + 4·25-s + 26-s + 5·27-s + 28-s + 8·29-s + 3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.14·19-s − 0.670·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.962·27-s + 0.188·28-s + 1.48·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38962 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38962 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.743772826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743772826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−14.59649244899494, −14.54552154918002, −13.70170753250588, −13.40312249097136, −12.42375299542250, −12.14732449893604, −11.76408920609584, −11.20875461594622, −11.05812573708181, −10.16443299584777, −9.724439732090118, −8.674824327279704, −8.303925305297349, −7.801760926582490, −7.249807651512219, −6.547211176239658, −6.085171015387120, −5.285144402681361, −4.941845730602801, −4.272765569829141, −3.689242793412410, −2.998213085519391, −2.521327184025247, −1.139485649136652, −0.6659234432785490,
0.6659234432785490, 1.139485649136652, 2.521327184025247, 2.998213085519391, 3.689242793412410, 4.272765569829141, 4.941845730602801, 5.285144402681361, 6.085171015387120, 6.547211176239658, 7.249807651512219, 7.801760926582490, 8.303925305297349, 8.674824327279704, 9.724439732090118, 10.16443299584777, 11.05812573708181, 11.20875461594622, 11.76408920609584, 12.14732449893604, 12.42375299542250, 13.40312249097136, 13.70170753250588, 14.54552154918002, 14.59649244899494
