| L(s) = 1 | + 3·3-s − 13.1·5-s + 15.6·7-s + 9·9-s − 55.7·11-s + 13·13-s − 39.3·15-s + 23.4·17-s − 25.6·19-s + 46.8·21-s − 189.·23-s + 47.2·25-s + 27·27-s − 236.·29-s + 47.0·31-s − 167.·33-s − 204.·35-s − 154.·37-s + 39·39-s − 34.6·41-s − 398.·43-s − 118.·45-s + 582.·47-s − 99.1·49-s + 70.4·51-s − 361.·53-s + 731.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.17·5-s + 0.843·7-s + 0.333·9-s − 1.52·11-s + 0.277·13-s − 0.677·15-s + 0.334·17-s − 0.309·19-s + 0.486·21-s − 1.71·23-s + 0.377·25-s + 0.192·27-s − 1.51·29-s + 0.272·31-s − 0.881·33-s − 0.989·35-s − 0.687·37-s + 0.160·39-s − 0.132·41-s − 1.41·43-s − 0.391·45-s + 1.80·47-s − 0.289·49-s + 0.193·51-s − 0.935·53-s + 1.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 13.1T + 125T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 + 55.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 23.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 34.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 396.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 211.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 85.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 651.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 927.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 391.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 745.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 173.T + 9.12e5T^{2} \) |
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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
−10.83394782654989645206140862298, −9.930619319756160097953807829626, −8.486485898348483876344336995153, −7.985894070032192047860297427511, −7.32760487574813187890454752491, −5.63734182546104978125567275658, −4.46759038480210206541089976156, −3.45709916490424119660510429234, −2.00838632588726881683853527300, 0,
2.00838632588726881683853527300, 3.45709916490424119660510429234, 4.46759038480210206541089976156, 5.63734182546104978125567275658, 7.32760487574813187890454752491, 7.985894070032192047860297427511, 8.486485898348483876344336995153, 9.930619319756160097953807829626, 10.83394782654989645206140862298
