| L(s) = 1 | − 25.6·3-s − 61.4·5-s + 184.·7-s + 415.·9-s + 130.·11-s − 1.17e3·13-s + 1.57e3·15-s − 493.·17-s − 2.42e3·19-s − 4.74e3·21-s + 4.18e3·23-s + 648.·25-s − 4.42e3·27-s + 1.39e3·29-s − 3.16e3·31-s − 3.34e3·33-s − 1.13e4·35-s + 1.29e4·37-s + 3.02e4·39-s + 9.52e3·41-s + 1.84e3·43-s − 2.55e4·45-s + 9.42e3·47-s + 1.73e4·49-s + 1.26e4·51-s − 5.07e3·53-s − 8.00e3·55-s + ⋯ |
| L(s) = 1 | − 1.64·3-s − 1.09·5-s + 1.42·7-s + 1.70·9-s + 0.324·11-s − 1.93·13-s + 1.80·15-s − 0.413·17-s − 1.54·19-s − 2.34·21-s + 1.64·23-s + 0.207·25-s − 1.16·27-s + 0.307·29-s − 0.591·31-s − 0.534·33-s − 1.56·35-s + 1.55·37-s + 3.18·39-s + 0.885·41-s + 0.152·43-s − 1.87·45-s + 0.622·47-s + 1.03·49-s + 0.681·51-s − 0.248·53-s − 0.357·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 3 | \( 1 + 25.6T + 243T^{2} \) |
| 5 | \( 1 + 61.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 184.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 130.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 493.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.29e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.52e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 9.42e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.07e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.99e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.14e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.66e4T + 8.58e9T^{2} \) |
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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
−9.364805234267249834628230115920, −8.175620863372611410086152250428, −7.38212645541471005938059868978, −6.70583123775209906406840698210, −5.46187945151400999887294663051, −4.64723843677961490554146105865, −4.29089110157867462122921737864, −2.31277437372406976350534316566, −0.917910839199161889213004194528, 0,
0.917910839199161889213004194528, 2.31277437372406976350534316566, 4.29089110157867462122921737864, 4.64723843677961490554146105865, 5.46187945151400999887294663051, 6.70583123775209906406840698210, 7.38212645541471005938059868978, 8.175620863372611410086152250428, 9.364805234267249834628230115920
