| L(s) = 1 | − 7.87·2-s − 28.4·3-s + 30.0·4-s − 12.9·5-s + 224.·6-s + 15.4·8-s + 566.·9-s + 101.·10-s + 291.·11-s − 854.·12-s + 169·13-s + 368.·15-s − 1.08e3·16-s + 1.57e3·17-s − 4.46e3·18-s − 2.56e3·19-s − 388.·20-s − 2.29e3·22-s − 1.10e3·23-s − 438.·24-s − 2.95e3·25-s − 1.33e3·26-s − 9.20e3·27-s + 8.09e3·29-s − 2.90e3·30-s − 3.32e3·31-s + 8.03e3·32-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 1.82·3-s + 0.938·4-s − 0.231·5-s + 2.54·6-s + 0.0851·8-s + 2.33·9-s + 0.322·10-s + 0.726·11-s − 1.71·12-s + 0.277·13-s + 0.422·15-s − 1.05·16-s + 1.32·17-s − 3.24·18-s − 1.63·19-s − 0.217·20-s − 1.01·22-s − 0.436·23-s − 0.155·24-s − 0.946·25-s − 0.386·26-s − 2.43·27-s + 1.78·29-s − 0.588·30-s − 0.621·31-s + 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 7.87T + 32T^{2} \) |
| 3 | \( 1 + 28.4T + 243T^{2} \) |
| 5 | \( 1 + 12.9T + 3.12e3T^{2} \) |
| 11 | \( 1 - 291.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.57e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 287.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.66e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.50e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e4T + 8.58e9T^{2} \) |
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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
−9.635557746934274562914261516637, −8.509283038623521019430673305630, −7.64435645535793706293725848515, −6.65268043506333357242046598370, −6.09889125114919453244431918172, −4.89566094555807432350796400849, −3.94646854218895445909236039884, −1.76732806662481318121352595909, −0.875654951713524917965500554729, 0,
0.875654951713524917965500554729, 1.76732806662481318121352595909, 3.94646854218895445909236039884, 4.89566094555807432350796400849, 6.09889125114919453244431918172, 6.65268043506333357242046598370, 7.64435645535793706293725848515, 8.509283038623521019430673305630, 9.635557746934274562914261516637
