| L(s) = 1 | + 3·3-s + 6.95·5-s + 9·9-s − 43.9·11-s + 83.5·13-s + 20.8·15-s − 10.4·17-s + 4.27·19-s − 160.·23-s − 76.5·25-s + 27·27-s + 9.93·29-s − 133.·31-s − 131.·33-s − 357.·37-s + 250.·39-s + 127.·41-s − 343.·43-s + 62.6·45-s + 77.4·47-s − 31.3·51-s − 460.·53-s − 305.·55-s + 12.8·57-s + 272.·59-s + 51.3·61-s + 581.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.622·5-s + 0.333·9-s − 1.20·11-s + 1.78·13-s + 0.359·15-s − 0.148·17-s + 0.0516·19-s − 1.45·23-s − 0.612·25-s + 0.192·27-s + 0.0635·29-s − 0.774·31-s − 0.694·33-s − 1.58·37-s + 1.02·39-s + 0.484·41-s − 1.21·43-s + 0.207·45-s + 0.240·47-s − 0.0859·51-s − 1.19·53-s − 0.749·55-s + 0.0298·57-s + 0.601·59-s + 0.107·61-s + 1.10·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6.95T + 125T^{2} \) |
| 11 | \( 1 + 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.27T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.93T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 77.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 460.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 272.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 51.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 327.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 571.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 923.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 97.6T + 9.12e5T^{2} \) |
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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
−8.296415394716699407146706429337, −7.68485070059839949233223790658, −6.63447782544841362007988187425, −5.88417439212052389323744446981, −5.21277158964052701423457356170, −4.01224083911137744005391565121, −3.32446503282435461266908869954, −2.21890908793913834133819503056, −1.49978067305721876004197705330, 0,
1.49978067305721876004197705330, 2.21890908793913834133819503056, 3.32446503282435461266908869954, 4.01224083911137744005391565121, 5.21277158964052701423457356170, 5.88417439212052389323744446981, 6.63447782544841362007988187425, 7.68485070059839949233223790658, 8.296415394716699407146706429337
