| L(s) = 1 | + 2.61·2-s + 4.85·4-s − 5-s − 7-s + 7.47·8-s − 2.61·10-s + 4.23·13-s − 2.61·14-s + 9.85·16-s − 2.47·17-s + 5.47·19-s − 4.85·20-s + 0.472·23-s − 4·25-s + 11.0·26-s − 4.85·28-s + 6.23·29-s + 8.47·31-s + 10.8·32-s − 6.47·34-s + 35-s + 3.47·37-s + 14.3·38-s − 7.47·40-s − 4.47·41-s + 4·43-s + 1.23·46-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s − 0.447·5-s − 0.377·7-s + 2.64·8-s − 0.827·10-s + 1.17·13-s − 0.699·14-s + 2.46·16-s − 0.599·17-s + 1.25·19-s − 1.08·20-s + 0.0984·23-s − 0.800·25-s + 2.17·26-s − 0.917·28-s + 1.15·29-s + 1.52·31-s + 1.91·32-s − 1.10·34-s + 0.169·35-s + 0.570·37-s + 2.32·38-s − 1.18·40-s − 0.698·41-s + 0.609·43-s + 0.182·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.986311125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.986311125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 - 4.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 - 0.472T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
| 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
−7.70556721209690893106970000191, −6.78520936500969628881143923525, −6.33020505163478152044122650853, −5.78774729073140963319274121558, −4.84400658006040156991923130204, −4.41531190092191203063922812696, −3.48254962602654026283734873520, −3.16782857261458040449854942769, −2.18306623022011482760887167795, −1.04212598280575554553967687476,
1.04212598280575554553967687476, 2.18306623022011482760887167795, 3.16782857261458040449854942769, 3.48254962602654026283734873520, 4.41531190092191203063922812696, 4.84400658006040156991923130204, 5.78774729073140963319274121558, 6.33020505163478152044122650853, 6.78520936500969628881143923525, 7.70556721209690893106970000191
