| L(s) = 1 | + 1.18·2-s − 2.96·3-s − 0.601·4-s − 5-s − 3.50·6-s − 3.50·7-s − 3.07·8-s + 5.80·9-s − 1.18·10-s − 5.93·11-s + 1.78·12-s − 0.182·13-s − 4.14·14-s + 2.96·15-s − 2.43·16-s + 4.65·17-s + 6.86·18-s + 0.308·19-s + 0.601·20-s + 10.4·21-s − 7.01·22-s − 0.491·23-s + 9.12·24-s + 25-s − 0.216·26-s − 8.30·27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | + 0.836·2-s − 1.71·3-s − 0.300·4-s − 0.447·5-s − 1.43·6-s − 1.32·7-s − 1.08·8-s + 1.93·9-s − 0.373·10-s − 1.78·11-s + 0.514·12-s − 0.0506·13-s − 1.10·14-s + 0.765·15-s − 0.608·16-s + 1.12·17-s + 1.61·18-s + 0.0708·19-s + 0.134·20-s + 2.27·21-s − 1.49·22-s − 0.102·23-s + 1.86·24-s + 0.200·25-s − 0.0423·26-s − 1.59·27-s + 0.398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 1201 | \( 1 + T \) |
| good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 + 5.93T + 11T^{2} \) |
| 13 | \( 1 + 0.182T + 13T^{2} \) |
| 17 | \( 1 - 4.65T + 17T^{2} \) |
| 19 | \( 1 - 0.308T + 19T^{2} \) |
| 23 | \( 1 + 0.491T + 23T^{2} \) |
| 29 | \( 1 - 5.76T + 29T^{2} \) |
| 31 | \( 1 - 1.87T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 + 9.02T + 43T^{2} \) |
| 47 | \( 1 - 0.0534T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 1.35T + 67T^{2} \) |
| 71 | \( 1 - 1.15T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 5.87T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−7.55281475569698242741070861254, −6.61133791364338337509099785850, −6.14580893323897792627200595699, −5.44906872526499221986447560677, −5.02804514511081941073566552996, −4.30296569718885408289309948247, −3.37077116568064355339973028855, −2.71904973416071021456024295406, −0.77341256288003247958458518441, 0,
0.77341256288003247958458518441, 2.71904973416071021456024295406, 3.37077116568064355339973028855, 4.30296569718885408289309948247, 5.02804514511081941073566552996, 5.44906872526499221986447560677, 6.14580893323897792627200595699, 6.61133791364338337509099785850, 7.55281475569698242741070861254
