| L(s) = 1 | + 15.8·3-s + 21·5-s + 6.75·9-s + 625.·11-s + 206.·13-s + 331.·15-s − 1.06e3·17-s − 1.88e3·19-s − 3.71e3·23-s − 2.68e3·25-s − 3.73e3·27-s − 123.·29-s − 9.10e3·31-s + 9.89e3·33-s − 6.02e3·37-s + 3.26e3·39-s + 1.72e4·41-s − 5.40e3·43-s + 141.·45-s + 1.87e3·47-s − 1.67e4·51-s + 1.87e4·53-s + 1.31e4·55-s − 2.97e4·57-s − 2.53e3·59-s + 2.09e3·61-s + 4.34e3·65-s + ⋯ |
| L(s) = 1 | + 1.01·3-s + 0.375·5-s + 0.0277·9-s + 1.55·11-s + 0.339·13-s + 0.380·15-s − 0.890·17-s − 1.19·19-s − 1.46·23-s − 0.858·25-s − 0.985·27-s − 0.0273·29-s − 1.70·31-s + 1.58·33-s − 0.723·37-s + 0.344·39-s + 1.59·41-s − 0.445·43-s + 0.0104·45-s + 0.123·47-s − 0.903·51-s + 0.914·53-s + 0.585·55-s − 1.21·57-s − 0.0948·59-s + 0.0720·61-s + 0.127·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 15.8T + 243T^{2} \) |
| 5 | \( 1 - 21T + 3.12e3T^{2} \) |
| 11 | \( 1 - 625.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 206.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 123.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.87e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.53e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.09e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.15e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.97e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−9.020603047895447709242670686503, −8.512615103993238944498144141569, −7.49857766961981031664043958794, −6.45091968534930724929007694292, −5.77656332887029407432273674295, −4.16644116552568009735179520216, −3.71001219026607564604780707094, −2.29903308546960652693627936272, −1.67393007106454773658773141793, 0,
1.67393007106454773658773141793, 2.29903308546960652693627936272, 3.71001219026607564604780707094, 4.16644116552568009735179520216, 5.77656332887029407432273674295, 6.45091968534930724929007694292, 7.49857766961981031664043958794, 8.512615103993238944498144141569, 9.020603047895447709242670686503
