| L(s) = 1 | − 0.314·5-s + 7-s + 1.47·11-s − 1.79·13-s − 5.22·17-s + 6.22·19-s + 6.58·23-s − 4.90·25-s + 7.43·29-s − 4.43·31-s − 0.314·35-s − 2.79·37-s + 8.37·41-s − 0.207·43-s − 4.06·47-s + 49-s + 4·53-s − 0.464·55-s − 4.06·59-s − 1.67·61-s + 0.563·65-s − 13.9·67-s − 7.27·71-s + 14.6·73-s + 1.47·77-s + 1.46·79-s + 12.2·83-s + ⋯ |
| L(s) = 1 | − 0.140·5-s + 0.377·7-s + 0.445·11-s − 0.497·13-s − 1.26·17-s + 1.42·19-s + 1.37·23-s − 0.980·25-s + 1.38·29-s − 0.796·31-s − 0.0531·35-s − 0.459·37-s + 1.30·41-s − 0.0316·43-s − 0.592·47-s + 0.142·49-s + 0.549·53-s − 0.0626·55-s − 0.529·59-s − 0.214·61-s + 0.0698·65-s − 1.70·67-s − 0.862·71-s + 1.71·73-s + 0.168·77-s + 0.164·79-s + 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.988509103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988509103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 0.314T + 5T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 + 2.79T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 0.207T + 43T^{2} \) |
| 47 | \( 1 + 4.06T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 + 1.67T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 7.27T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.935181539008938167931869340948, −7.40814103085314761818357222117, −6.73948086887128507377384589038, −5.97383044705119987101182162874, −5.04258058462312142321834636916, −4.58429676106604564289113729353, −3.62541178626573742165541987047, −2.79514425384011553536546920670, −1.83896078671121413008254932303, −0.75178492193240672237652132633,
0.75178492193240672237652132633, 1.83896078671121413008254932303, 2.79514425384011553536546920670, 3.62541178626573742165541987047, 4.58429676106604564289113729353, 5.04258058462312142321834636916, 5.97383044705119987101182162874, 6.73948086887128507377384589038, 7.40814103085314761818357222117, 7.935181539008938167931869340948
