| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 6·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 12·10-s + 48·11-s + 12·12-s + 38·13-s − 14·14-s − 18·15-s + 16·16-s + 114·17-s − 18·18-s + 56·19-s − 24·20-s + 21·21-s − 96·22-s − 23·23-s − 24·24-s − 89·25-s − 76·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.536·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.379·10-s + 1.31·11-s + 0.288·12-s + 0.810·13-s − 0.267·14-s − 0.309·15-s + 1/4·16-s + 1.62·17-s − 0.235·18-s + 0.676·19-s − 0.268·20-s + 0.218·21-s − 0.930·22-s − 0.208·23-s − 0.204·24-s − 0.711·25-s − 0.573·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.151945164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.151945164\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 46 T + p^{3} T^{2} \) |
| 41 | \( 1 + 342 T + p^{3} T^{2} \) |
| 43 | \( 1 - 248 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 426 T + p^{3} T^{2} \) |
| 59 | \( 1 + 852 T + p^{3} T^{2} \) |
| 61 | \( 1 - 338 T + p^{3} T^{2} \) |
| 67 | \( 1 - 488 T + p^{3} T^{2} \) |
| 71 | \( 1 - 336 T + p^{3} T^{2} \) |
| 73 | \( 1 - 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1184 T + p^{3} T^{2} \) |
| 83 | \( 1 + 336 T + p^{3} T^{2} \) |
| 89 | \( 1 + 78 T + p^{3} T^{2} \) |
| 97 | \( 1 - 746 T + p^{3} T^{2} \) |
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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
−9.485093931194502059303510772883, −8.828815052651796549238247956986, −7.951253065652994457345263422807, −7.46627351171530597986557634906, −6.41745006592126048931574577775, −5.39963444467444650538077258482, −3.90148382635234953134373333084, −3.39062714554572102680928321995, −1.81266000355776379408262128622, −0.918270324329344189851945108512,
0.918270324329344189851945108512, 1.81266000355776379408262128622, 3.39062714554572102680928321995, 3.90148382635234953134373333084, 5.39963444467444650538077258482, 6.41745006592126048931574577775, 7.46627351171530597986557634906, 7.951253065652994457345263422807, 8.828815052651796549238247956986, 9.485093931194502059303510772883
