| 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
| \(L(1)\) |
\(\approx\) |
\(4.078177448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.078177448\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.037026095581285762206835280901, −7.09908487924829304872258037180, −6.27191357329978686093851598351, −5.85757142497446157230585250071, −5.22358847641312451893741652074, −4.70918236854112951118035413929, −3.59379910137062603414278825509, −2.79291815575222602254082312693, −2.28548884330437056312406294489, −0.903498625882440795641030404725,
0.903498625882440795641030404725, 2.28548884330437056312406294489, 2.79291815575222602254082312693, 3.59379910137062603414278825509, 4.70918236854112951118035413929, 5.22358847641312451893741652074, 5.85757142497446157230585250071, 6.27191357329978686093851598351, 7.09908487924829304872258037180, 8.037026095581285762206835280901
