| L(s) = 1 | + 5-s − 1.64·7-s − 1.64·11-s + 3.64·13-s + 4.64·17-s − 2.64·19-s − 4.29·23-s + 25-s + 5.64·29-s + 2.64·31-s − 1.64·35-s − 5.29·37-s + 0.354·41-s + 6.93·43-s + 5.29·47-s − 4.29·49-s − 3.93·53-s − 1.64·55-s + 3.64·59-s + 7.58·61-s + 3.64·65-s − 6·67-s − 2.35·71-s − 0.937·73-s + 2.70·77-s + 7.35·79-s + 5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.622·7-s − 0.496·11-s + 1.01·13-s + 1.12·17-s − 0.606·19-s − 0.894·23-s + 0.200·25-s + 1.04·29-s + 0.475·31-s − 0.278·35-s − 0.869·37-s + 0.0553·41-s + 1.05·43-s + 0.771·47-s − 0.613·49-s − 0.540·53-s − 0.221·55-s + 0.474·59-s + 0.970·61-s + 0.452·65-s − 0.733·67-s − 0.279·71-s − 0.109·73-s + 0.308·77-s + 0.827·79-s + 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.938477017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938477017\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 1.64T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.29T + 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 - 0.354T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 + 0.937T + 73T^{2} \) |
| 79 | \( 1 - 7.35T + 79T^{2} \) |
| 83 | \( 1 - 5T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 + 0.708T + 97T^{2} \) |
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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
−8.372872036062919883357839306801, −7.74778608497315369405367205073, −6.79133009161244472987006478730, −6.11999698141697022774408086191, −5.62096291883063601297755409610, −4.63933696490726698959996407556, −3.70345255716356488341126169930, −2.97095393789605299904172237625, −1.97306709825086016070659793986, −0.790560549381691483083584726871,
0.790560549381691483083584726871, 1.97306709825086016070659793986, 2.97095393789605299904172237625, 3.70345255716356488341126169930, 4.63933696490726698959996407556, 5.62096291883063601297755409610, 6.11999698141697022774408086191, 6.79133009161244472987006478730, 7.74778608497315369405367205073, 8.372872036062919883357839306801
