| L(s) = 1 | − 1.56·2-s + 0.436·4-s − 3.73·5-s − 1.31·7-s + 2.44·8-s + 5.82·10-s + 2.33·11-s − 1.43·13-s + 2.05·14-s − 4.68·16-s − 4.11·17-s + 1.75·19-s − 1.62·20-s − 3.64·22-s − 23-s + 8.92·25-s + 2.24·26-s − 0.575·28-s + 29-s + 8.00·31-s + 2.42·32-s + 6.43·34-s + 4.91·35-s + 3.69·37-s − 2.73·38-s − 9.10·40-s − 6.51·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.218·4-s − 1.66·5-s − 0.497·7-s + 0.862·8-s + 1.84·10-s + 0.703·11-s − 0.398·13-s + 0.549·14-s − 1.17·16-s − 0.999·17-s + 0.401·19-s − 0.364·20-s − 0.777·22-s − 0.208·23-s + 1.78·25-s + 0.439·26-s − 0.108·28-s + 0.185·29-s + 1.43·31-s + 0.429·32-s + 1.10·34-s + 0.830·35-s + 0.607·37-s − 0.443·38-s − 1.43·40-s − 1.01·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3054703296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3054703296\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 + 1.43T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 31 | \( 1 - 8.00T + 31T^{2} \) |
| 37 | \( 1 - 3.69T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 + 2.72T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 + 7.02T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
| 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
−8.091633128195415081382163471779, −7.64604991496881794766927273413, −6.84288728852347446606748179986, −6.40526005416421213660903624334, −4.89561755333589666935487793217, −4.42611914073541856161076311508, −3.67480199848175966925349408773, −2.79752625200573623743599068036, −1.45047315926808917513458483241, −0.36439457228597019299520947485,
0.36439457228597019299520947485, 1.45047315926808917513458483241, 2.79752625200573623743599068036, 3.67480199848175966925349408773, 4.42611914073541856161076311508, 4.89561755333589666935487793217, 6.40526005416421213660903624334, 6.84288728852347446606748179986, 7.64604991496881794766927273413, 8.091633128195415081382163471779
