| L(s) = 1 | + 3·3-s − 5·5-s − 22.2·7-s + 9·9-s − 55.6·11-s + 13·13-s − 15·15-s − 95.2·17-s + 43.8·19-s − 66.7·21-s − 89.2·23-s + 25·25-s + 27·27-s − 46.5·29-s + 60.6·31-s − 167.·33-s + 111.·35-s + 122.·37-s + 39·39-s − 23.8·41-s + 287.·43-s − 45·45-s + 249.·47-s + 152.·49-s − 285.·51-s − 267.·53-s + 278.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.20·7-s + 0.333·9-s − 1.52·11-s + 0.277·13-s − 0.258·15-s − 1.35·17-s + 0.529·19-s − 0.694·21-s − 0.809·23-s + 0.200·25-s + 0.192·27-s − 0.297·29-s + 0.351·31-s − 0.881·33-s + 0.537·35-s + 0.545·37-s + 0.160·39-s − 0.0910·41-s + 1.02·43-s − 0.149·45-s + 0.774·47-s + 0.445·49-s − 0.784·51-s − 0.694·53-s + 0.682·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.151200048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.151200048\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 + 22.2T + 343T^{2} \) |
| 11 | \( 1 + 55.6T + 1.33e3T^{2} \) |
| 17 | \( 1 + 95.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 89.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 46.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 60.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 23.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 267.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 553.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 582.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 806.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 105.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 63.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 8.82T + 7.04e5T^{2} \) |
| 97 | \( 1 - 116.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.142360509259237081808756684381, −8.181068106732297134149391398636, −7.62032707579900983705201457450, −6.71299665495027736941443166157, −5.91191496243546066276797009003, −4.80288466265815051817659117140, −3.85364668142706608520937908920, −2.97737464927210586486172842067, −2.22031347270113825194863884767, −0.47435760369552245064640660973,
0.47435760369552245064640660973, 2.22031347270113825194863884767, 2.97737464927210586486172842067, 3.85364668142706608520937908920, 4.80288466265815051817659117140, 5.91191496243546066276797009003, 6.71299665495027736941443166157, 7.62032707579900983705201457450, 8.181068106732297134149391398636, 9.142360509259237081808756684381
