| L(s) = 1 | − 2-s − 1.98·3-s + 4-s − 5-s + 1.98·6-s + 4.00·7-s − 8-s + 0.947·9-s + 10-s + 5.57·11-s − 1.98·12-s − 0.0521·13-s − 4.00·14-s + 1.98·15-s + 16-s + 5.25·17-s − 0.947·18-s − 8.35·19-s − 20-s − 7.95·21-s − 5.57·22-s − 4.82·23-s + 1.98·24-s + 25-s + 0.0521·26-s + 4.07·27-s + 4.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.14·3-s + 0.5·4-s − 0.447·5-s + 0.811·6-s + 1.51·7-s − 0.353·8-s + 0.315·9-s + 0.316·10-s + 1.68·11-s − 0.573·12-s − 0.0144·13-s − 1.07·14-s + 0.513·15-s + 0.250·16-s + 1.27·17-s − 0.223·18-s − 1.91·19-s − 0.223·20-s − 1.73·21-s − 1.18·22-s − 1.00·23-s + 0.405·24-s + 0.200·25-s + 0.0102·26-s + 0.784·27-s + 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 7 | \( 1 - 4.00T + 7T^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 + 0.0521T + 13T^{2} \) |
| 17 | \( 1 - 5.25T + 17T^{2} \) |
| 19 | \( 1 + 8.35T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 3.07T + 29T^{2} \) |
| 31 | \( 1 - 7.18T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.64T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.07T + 59T^{2} \) |
| 61 | \( 1 + 9.17T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 0.379T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 + 4.02T + 97T^{2} \) |
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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
−7.88438725421720333125609938368, −6.94097239056862116641316342778, −6.34411386992327035458653938233, −5.78663621363092757918506165991, −4.73318249416285677970219678303, −4.34846918876593614159106111391, −3.24806859287944530758618363535, −1.77921635894065114106837479700, −1.25983485758310745453581000061, 0,
1.25983485758310745453581000061, 1.77921635894065114106837479700, 3.24806859287944530758618363535, 4.34846918876593614159106111391, 4.73318249416285677970219678303, 5.78663621363092757918506165991, 6.34411386992327035458653938233, 6.94097239056862116641316342778, 7.88438725421720333125609938368
