| L(s) = 1 | − 2.80·2-s + 5.86·4-s − 2.52·5-s − 2.67·7-s − 10.8·8-s + 7.08·10-s + 0.714·11-s + 0.530·13-s + 7.48·14-s + 18.6·16-s + 8.04·17-s − 5.87·19-s − 14.8·20-s − 2.00·22-s − 23-s + 1.39·25-s − 1.48·26-s − 15.6·28-s − 29-s − 0.921·31-s − 30.5·32-s − 22.5·34-s + 6.75·35-s − 0.783·37-s + 16.4·38-s + 27.3·40-s + 4.70·41-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 2.93·4-s − 1.13·5-s − 1.00·7-s − 3.82·8-s + 2.24·10-s + 0.215·11-s + 0.147·13-s + 2.00·14-s + 4.65·16-s + 1.95·17-s − 1.34·19-s − 3.31·20-s − 0.427·22-s − 0.208·23-s + 0.278·25-s − 0.291·26-s − 2.95·28-s − 0.185·29-s − 0.165·31-s − 5.40·32-s − 3.87·34-s + 1.14·35-s − 0.128·37-s + 2.67·38-s + 4.32·40-s + 0.734·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3272521277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3272521277\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.80T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 0.714T + 11T^{2} \) |
| 13 | \( 1 - 0.530T + 13T^{2} \) |
| 17 | \( 1 - 8.04T + 17T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 31 | \( 1 + 0.921T + 31T^{2} \) |
| 37 | \( 1 + 0.783T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 1.19T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 6.08T + 73T^{2} \) |
| 79 | \( 1 + 3.02T + 79T^{2} \) |
| 83 | \( 1 + 1.50T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.960006395046583319591501532705, −7.67299047569231427685166169600, −7.04802364168102498628364682908, −6.18887351435171148677453011187, −5.76735253623441202893979613764, −4.08493961295472793169205250629, −3.33459042782096730081068617509, −2.63792701702039013946764828955, −1.42415395973067725934490430275, −0.42287126829944839345672845165,
0.42287126829944839345672845165, 1.42415395973067725934490430275, 2.63792701702039013946764828955, 3.33459042782096730081068617509, 4.08493961295472793169205250629, 5.76735253623441202893979613764, 6.18887351435171148677453011187, 7.04802364168102498628364682908, 7.67299047569231427685166169600, 7.960006395046583319591501532705
