| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 2.76·7-s − 8·8-s − 10·10-s − 7.70·11-s − 15.9·13-s + 5.52·14-s + 16·16-s − 44.0·17-s − 19·19-s + 20·20-s + 15.4·22-s + 82.2·23-s + 25·25-s + 31.9·26-s − 11.0·28-s + 72.6·29-s − 219.·31-s − 32·32-s + 88.0·34-s − 13.8·35-s − 362.·37-s + 38·38-s − 40·40-s + 150.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.149·7-s − 0.353·8-s − 0.316·10-s − 0.211·11-s − 0.340·13-s + 0.105·14-s + 0.250·16-s − 0.628·17-s − 0.229·19-s + 0.223·20-s + 0.149·22-s + 0.745·23-s + 0.200·25-s + 0.240·26-s − 0.0746·28-s + 0.465·29-s − 1.26·31-s − 0.176·32-s + 0.444·34-s − 0.0667·35-s − 1.61·37-s + 0.162·38-s − 0.158·40-s + 0.571·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.309323872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309323872\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 19 | \( 1 + 19T \) |
| good | 7 | \( 1 + 2.76T + 343T^{2} \) |
| 11 | \( 1 + 7.70T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 82.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 72.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 362.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 331.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 267.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 759.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 721.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 664.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 437.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 994.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 235.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.16e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−8.900497066926777244821497121989, −8.433883189384188871906007728657, −7.25448688633644984031594585905, −6.84702631283612892096429674656, −5.79439572413839987711811246266, −5.03548211403267582836477591172, −3.82717252843613819806227950621, −2.68893649975552595048461067504, −1.85447858055978662767160817768, −0.59405021345018595691920316983,
0.59405021345018595691920316983, 1.85447858055978662767160817768, 2.68893649975552595048461067504, 3.82717252843613819806227950621, 5.03548211403267582836477591172, 5.79439572413839987711811246266, 6.84702631283612892096429674656, 7.25448688633644984031594585905, 8.433883189384188871906007728657, 8.900497066926777244821497121989
