| L(s) = 1 | − 2.35·2-s − 2.44·4-s + 8.17·5-s + 34.4·7-s + 24.6·8-s − 19.2·10-s + 20.0·11-s + 4.61·13-s − 81.2·14-s − 38.4·16-s + 39.0·17-s + 72.4·19-s − 19.9·20-s − 47.3·22-s + 183.·23-s − 58.2·25-s − 10.8·26-s − 84.1·28-s + 67.6·29-s + 261.·31-s − 106.·32-s − 92.1·34-s + 281.·35-s + 260.·37-s − 170.·38-s + 201.·40-s − 90.3·41-s + ⋯ |
| L(s) = 1 | − 0.833·2-s − 0.305·4-s + 0.730·5-s + 1.86·7-s + 1.08·8-s − 0.609·10-s + 0.550·11-s + 0.0984·13-s − 1.55·14-s − 0.601·16-s + 0.557·17-s + 0.874·19-s − 0.223·20-s − 0.459·22-s + 1.66·23-s − 0.465·25-s − 0.0820·26-s − 0.568·28-s + 0.432·29-s + 1.51·31-s − 0.586·32-s − 0.464·34-s + 1.35·35-s + 1.15·37-s − 0.729·38-s + 0.795·40-s − 0.344·41-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
| \(L(2)\) |
\(\approx\) |
\(2.512431189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.512431189\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.35T + 8T^{2} \) |
| 5 | \( 1 - 8.17T + 125T^{2} \) |
| 7 | \( 1 - 34.4T + 343T^{2} \) |
| 11 | \( 1 - 20.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.61T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 67.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 261.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 90.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 612.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 592.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 641.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 76.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 100.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 900.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 651.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 856.T + 9.12e5T^{2} \) |
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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
−8.601930135769348927591420947243, −8.163968706363070341201765006713, −7.47816765617182152199276639805, −6.52359859844255242334619247445, −5.26837138754897703287643390029, −4.95013755430781708991855416963, −3.96139960394576254590622969054, −2.51625466447687360475659986500, −1.33342471258703073450467620280, −1.03286909423517146993236632552,
1.03286909423517146993236632552, 1.33342471258703073450467620280, 2.51625466447687360475659986500, 3.96139960394576254590622969054, 4.95013755430781708991855416963, 5.26837138754897703287643390029, 6.52359859844255242334619247445, 7.47816765617182152199276639805, 8.163968706363070341201765006713, 8.601930135769348927591420947243
