| L(s) = 1 | + 1.82·2-s − 4.65·4-s + 21.6·5-s − 18.0·7-s − 23.1·8-s + 39.5·10-s + 37.0·11-s + 42.5·13-s − 33.0·14-s − 5.00·16-s + 15.1·17-s + 65.0·19-s − 100.·20-s + 67.6·22-s − 131.·23-s + 343.·25-s + 77.8·26-s + 84.2·28-s + 234.·29-s − 326.·31-s + 175.·32-s + 27.7·34-s − 391.·35-s + 374.·37-s + 118.·38-s − 500.·40-s − 47.6·41-s + ⋯ |
| L(s) = 1 | + 0.646·2-s − 0.582·4-s + 1.93·5-s − 0.975·7-s − 1.02·8-s + 1.25·10-s + 1.01·11-s + 0.908·13-s − 0.630·14-s − 0.0782·16-s + 0.216·17-s + 0.785·19-s − 1.12·20-s + 0.655·22-s − 1.18·23-s + 2.74·25-s + 0.586·26-s + 0.568·28-s + 1.50·29-s − 1.89·31-s + 0.971·32-s + 0.139·34-s − 1.88·35-s + 1.66·37-s + 0.507·38-s − 1.97·40-s − 0.181·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.844400246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.844400246\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 1.82T + 8T^{2} \) |
| 5 | \( 1 - 21.6T + 125T^{2} \) |
| 7 | \( 1 + 18.0T + 343T^{2} \) |
| 11 | \( 1 - 37.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 15.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 326.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 374.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 47.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 7.23T + 1.48e5T^{2} \) |
| 59 | \( 1 + 747.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 284.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 505.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 802.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 443.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 453.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 649.T + 9.12e5T^{2} \) |
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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
−9.108931299573965142803118617220, −8.108599971166133941348645022779, −6.45759662858852995822980771970, −6.41038525839456851115301251370, −5.63301927023214175885872658935, −4.86135710108654599625955714628, −3.71347142480627623314998469264, −3.06778854781970457900470346883, −1.86997268517030649952779948155, −0.839234527531099334663943457325,
0.839234527531099334663943457325, 1.86997268517030649952779948155, 3.06778854781970457900470346883, 3.71347142480627623314998469264, 4.86135710108654599625955714628, 5.63301927023214175885872658935, 6.41038525839456851115301251370, 6.45759662858852995822980771970, 8.108599971166133941348645022779, 9.108931299573965142803118617220
