| L(s) = 1 | + 3.46·5-s + 7-s + 3.46·11-s + 5·13-s + 19-s + 6.92·23-s + 6.99·25-s − 3.46·29-s − 5·31-s + 3.46·35-s − 37-s + 3.46·41-s + 43-s + 3.46·47-s − 6·49-s − 10.3·53-s + 11.9·55-s − 3.46·59-s + 2·61-s + 17.3·65-s − 8·67-s − 10.3·71-s + 2·73-s + 3.46·77-s + 79-s − 6.92·83-s + 10.3·89-s + ⋯ |
| L(s) = 1 | + 1.54·5-s + 0.377·7-s + 1.04·11-s + 1.38·13-s + 0.229·19-s + 1.44·23-s + 1.39·25-s − 0.643·29-s − 0.898·31-s + 0.585·35-s − 0.164·37-s + 0.541·41-s + 0.152·43-s + 0.505·47-s − 0.857·49-s − 1.42·53-s + 1.61·55-s − 0.450·59-s + 0.256·61-s + 2.14·65-s − 0.977·67-s − 1.23·71-s + 0.234·73-s + 0.394·77-s + 0.112·79-s − 0.760·83-s + 1.10·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.302387758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.302387758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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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
−8.823148884472770583686001033090, −7.72253562023634107256833095808, −6.81574589779543637368539971185, −6.19601898208036417515526512565, −5.63429678017844926628142228106, −4.83706904053187060669113718097, −3.81573367725641187139774016798, −2.92573440025065869437037121154, −1.72304635608116323224123903823, −1.23579631315049212244451551519,
1.23579631315049212244451551519, 1.72304635608116323224123903823, 2.92573440025065869437037121154, 3.81573367725641187139774016798, 4.83706904053187060669113718097, 5.63429678017844926628142228106, 6.19601898208036417515526512565, 6.81574589779543637368539971185, 7.72253562023634107256833095808, 8.823148884472770583686001033090
