| L(s) = 1 | − 5.02·2-s + 17.2·4-s − 20.4·5-s − 7·7-s − 46.2·8-s + 102.·10-s + 11·11-s − 0.0115·13-s + 35.1·14-s + 94.7·16-s + 9.52·17-s − 93.4·19-s − 351.·20-s − 55.2·22-s − 99.9·23-s + 292.·25-s + 0.0580·26-s − 120.·28-s − 276.·29-s − 181.·31-s − 105.·32-s − 47.8·34-s + 143.·35-s − 404.·37-s + 469.·38-s + 946.·40-s + 27.8·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.15·4-s − 1.82·5-s − 0.377·7-s − 2.04·8-s + 3.24·10-s + 0.301·11-s − 0.000246·13-s + 0.671·14-s + 1.47·16-s + 0.135·17-s − 1.12·19-s − 3.93·20-s − 0.535·22-s − 0.906·23-s + 2.34·25-s + 0.000437·26-s − 0.813·28-s − 1.76·29-s − 1.05·31-s − 0.581·32-s − 0.241·34-s + 0.690·35-s − 1.79·37-s + 2.00·38-s + 3.73·40-s + 0.105·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1136545527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1136545527\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 5.02T + 8T^{2} \) |
| 5 | \( 1 + 20.4T + 125T^{2} \) |
| 13 | \( 1 + 0.0115T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.52T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 404.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 27.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 76.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 170.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 585.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 530.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 354.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 471.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 563.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 895.T + 9.12e5T^{2} \) |
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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
−10.04426306200261277331651621119, −8.857511806299228346777237348319, −8.579255538198677495283379369288, −7.39116727870224552855130335791, −7.28848752239255272817711067074, −6.00394131974767877998440514487, −4.23839169270983097455629963554, −3.30401790577683581397018874534, −1.79039089111195507522466622202, −0.24614945773533279077487588749,
0.24614945773533279077487588749, 1.79039089111195507522466622202, 3.30401790577683581397018874534, 4.23839169270983097455629963554, 6.00394131974767877998440514487, 7.28848752239255272817711067074, 7.39116727870224552855130335791, 8.579255538198677495283379369288, 8.857511806299228346777237348319, 10.04426306200261277331651621119
