| L(s) = 1 | + 4·2-s + 16·4-s + 21·5-s + 74·7-s + 64·8-s + 84·10-s + 270·11-s − 115·13-s + 296·14-s + 256·16-s − 861·17-s + 1.85e3·19-s + 336·20-s + 1.08e3·22-s + 3.61e3·23-s − 2.68e3·25-s − 460·26-s + 1.18e3·28-s + 1.12e3·29-s + 5.22e3·31-s + 1.02e3·32-s − 3.44e3·34-s + 1.55e3·35-s + 9.91e3·37-s + 7.40e3·38-s + 1.34e3·40-s + 1.07e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.375·5-s + 0.570·7-s + 0.353·8-s + 0.265·10-s + 0.672·11-s − 0.188·13-s + 0.403·14-s + 1/4·16-s − 0.722·17-s + 1.17·19-s + 0.187·20-s + 0.475·22-s + 1.42·23-s − 0.858·25-s − 0.133·26-s + 0.285·28-s + 0.248·29-s + 0.977·31-s + 0.176·32-s − 0.510·34-s + 0.214·35-s + 1.19·37-s + 0.831·38-s + 0.132·40-s + 0.999·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.688160762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.688160762\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 21 T + p^{5} T^{2} \) |
| 7 | \( 1 - 74 T + p^{5} T^{2} \) |
| 11 | \( 1 - 270 T + p^{5} T^{2} \) |
| 13 | \( 1 + 115 T + p^{5} T^{2} \) |
| 17 | \( 1 + 861 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1850 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3618 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1125 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5228 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9917 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10758 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19714 T + p^{5} T^{2} \) |
| 47 | \( 1 - 9984 T + p^{5} T^{2} \) |
| 53 | \( 1 - 36726 T + p^{5} T^{2} \) |
| 59 | \( 1 - 26460 T + p^{5} T^{2} \) |
| 61 | \( 1 + 53779 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12934 T + p^{5} T^{2} \) |
| 71 | \( 1 + 4254 T + p^{5} T^{2} \) |
| 73 | \( 1 + 17521 T + p^{5} T^{2} \) |
| 79 | \( 1 + 36946 T + p^{5} T^{2} \) |
| 83 | \( 1 + 76416 T + p^{5} T^{2} \) |
| 89 | \( 1 + 45357 T + p^{5} T^{2} \) |
| 97 | \( 1 - 127574 T + p^{5} T^{2} \) |
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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
−11.86939120401867914331251600869, −11.28350220388998770695686811362, −10.02002315784318190620953317008, −8.925599390208910191938525609957, −7.58855218103548343461047267106, −6.50092836864793372121601640969, −5.32492397334395679959911029020, −4.25849581621773065977264689695, −2.75100931142022291058618139198, −1.27017748816807925944253622820,
1.27017748816807925944253622820, 2.75100931142022291058618139198, 4.25849581621773065977264689695, 5.32492397334395679959911029020, 6.50092836864793372121601640969, 7.58855218103548343461047267106, 8.925599390208910191938525609957, 10.02002315784318190620953317008, 11.28350220388998770695686811362, 11.86939120401867914331251600869
