| L(s) = 1 | + 2.44·2-s + 3-s + 3.97·4-s − 3.84·5-s + 2.44·6-s + 7-s + 4.82·8-s + 9-s − 9.38·10-s − 3.40·11-s + 3.97·12-s + 2.24·13-s + 2.44·14-s − 3.84·15-s + 3.85·16-s + 2.44·18-s + 5.37·19-s − 15.2·20-s + 21-s − 8.32·22-s + 6.36·23-s + 4.82·24-s + 9.75·25-s + 5.48·26-s + 27-s + 3.97·28-s − 2.57·29-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.577·3-s + 1.98·4-s − 1.71·5-s + 0.997·6-s + 0.377·7-s + 1.70·8-s + 0.333·9-s − 2.96·10-s − 1.02·11-s + 1.14·12-s + 0.622·13-s + 0.653·14-s − 0.991·15-s + 0.963·16-s + 0.576·18-s + 1.23·19-s − 3.41·20-s + 0.218·21-s − 1.77·22-s + 1.32·23-s + 0.985·24-s + 1.95·25-s + 1.07·26-s + 0.192·27-s + 0.751·28-s − 0.478·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.736819405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.736819405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 - 6.15T + 31T^{2} \) |
| 37 | \( 1 - 2.65T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 - 6.35T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 5.44T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 5.08T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
| 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
−7.907708298459489959057220676123, −7.26188281477385316473060092282, −6.75106518419461479117793701972, −5.60061715284642230087523994471, −4.98272954870835805217996256001, −4.38852418553877112808206776337, −3.64118159732722030994095157368, −3.15482433615904868595599602705, −2.44552822827112690301115825326, −0.961198156548361789416518727724,
0.961198156548361789416518727724, 2.44552822827112690301115825326, 3.15482433615904868595599602705, 3.64118159732722030994095157368, 4.38852418553877112808206776337, 4.98272954870835805217996256001, 5.60061715284642230087523994471, 6.75106518419461479117793701972, 7.26188281477385316473060092282, 7.907708298459489959057220676123
