| L(s) = 1 | + 7.21·3-s − 5·5-s + 7.21·7-s + 24.9·9-s + 43.2·11-s + 34·13-s − 36.0·15-s + 114·17-s + 51.9·21-s − 209.·23-s + 25·25-s − 14.4·27-s − 26·29-s + 100.·31-s + 312·33-s − 36.0·35-s − 150·37-s + 245.·39-s + 342·41-s − 454.·43-s − 124.·45-s + 584.·47-s − 291·49-s + 822.·51-s − 262·53-s − 216.·55-s − 490.·59-s + ⋯ |
| L(s) = 1 | + 1.38·3-s − 0.447·5-s + 0.389·7-s + 0.925·9-s + 1.18·11-s + 0.725·13-s − 0.620·15-s + 1.62·17-s + 0.540·21-s − 1.89·23-s + 0.200·25-s − 0.102·27-s − 0.166·29-s + 0.584·31-s + 1.64·33-s − 0.174·35-s − 0.666·37-s + 1.00·39-s + 1.30·41-s − 1.61·43-s − 0.414·45-s + 1.81·47-s − 0.848·49-s + 2.25·51-s − 0.679·53-s − 0.530·55-s − 1.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.708118460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708118460\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 - 7.21T + 27T^{2} \) |
| 7 | \( 1 - 7.21T + 343T^{2} \) |
| 11 | \( 1 - 43.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34T + 2.19e3T^{2} \) |
| 17 | \( 1 - 114T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 209.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 26T + 2.43e4T^{2} \) |
| 31 | \( 1 - 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150T + 5.06e4T^{2} \) |
| 41 | \( 1 - 342T + 6.89e4T^{2} \) |
| 43 | \( 1 + 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 584.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 262T + 1.48e5T^{2} \) |
| 59 | \( 1 + 490.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 262T + 2.26e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 682T + 3.89e5T^{2} \) |
| 79 | \( 1 + 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 151.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 + 966T + 9.12e5T^{2} \) |
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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
−12.38487837713366474078240689695, −11.59891154421004489653131496834, −10.18308817016433810082649079818, −9.172901688584562362713320968989, −8.257862174868890162050513331404, −7.56961178871412032587388056431, −6.03436319431682710364545563810, −4.15829131094677255206442025173, −3.27458454718604352225616135221, −1.56007874739793814512359953671,
1.56007874739793814512359953671, 3.27458454718604352225616135221, 4.15829131094677255206442025173, 6.03436319431682710364545563810, 7.56961178871412032587388056431, 8.257862174868890162050513331404, 9.172901688584562362713320968989, 10.18308817016433810082649079818, 11.59891154421004489653131496834, 12.38487837713366474078240689695
