| L(s) = 1 | + 1.41·2-s + 0.00250·4-s − 0.0661·5-s + 1.68·7-s − 2.82·8-s − 0.0936·10-s + 6.36·11-s + 1.48·13-s + 2.38·14-s − 4.00·16-s + 0.164·17-s + 4.96·19-s − 0.000165·20-s + 9.00·22-s − 23-s − 4.99·25-s + 2.09·26-s + 0.00421·28-s − 29-s + 3.38·31-s − 0.0141·32-s + 0.233·34-s − 0.111·35-s + 10.0·37-s + 7.01·38-s + 0.186·40-s − 7.66·41-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.00125·4-s − 0.0295·5-s + 0.636·7-s − 0.999·8-s − 0.0296·10-s + 1.91·11-s + 0.411·13-s + 0.636·14-s − 1.00·16-s + 0.0400·17-s + 1.13·19-s − 3.70e − 5·20-s + 1.91·22-s − 0.208·23-s − 0.999·25-s + 0.411·26-s + 0.000795·28-s − 0.185·29-s + 0.608·31-s − 0.00250·32-s + 0.0400·34-s − 0.0188·35-s + 1.64·37-s + 1.13·38-s + 0.0295·40-s − 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.693950023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.693950023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 0.0661T + 5T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 - 6.36T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 - 0.164T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 7.66T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 - 7.72T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 0.107T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 + 8.12T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−8.113170023499113855211387265821, −7.24100738205086579806526028471, −6.34065589472456880317060071499, −5.97805876654207761174997435492, −5.06441874218997590853109955772, −4.39990107988981554451537340942, −3.78297815530010231846581064659, −3.15827407985441370163004942134, −1.89631197407450844602335007918, −0.925104740438308398245953292317,
0.925104740438308398245953292317, 1.89631197407450844602335007918, 3.15827407985441370163004942134, 3.78297815530010231846581064659, 4.39990107988981554451537340942, 5.06441874218997590853109955772, 5.97805876654207761174997435492, 6.34065589472456880317060071499, 7.24100738205086579806526028471, 8.113170023499113855211387265821
