| L(s) = 1 | − 0.442·3-s − 0.891·5-s − 2.52·7-s − 2.80·9-s − 1.95·11-s + 6.45·13-s + 0.394·15-s − 3.42·17-s + 1.11·21-s − 8.18·23-s − 4.20·25-s + 2.56·27-s − 4.58·29-s − 8.79·31-s + 0.863·33-s + 2.24·35-s + 5.97·37-s − 2.85·39-s + 3.48·41-s + 6.24·43-s + 2.49·45-s + 10.5·47-s − 0.648·49-s + 1.51·51-s − 3.76·53-s + 1.74·55-s + 2.84·59-s + ⋯ |
| L(s) = 1 | − 0.255·3-s − 0.398·5-s − 0.952·7-s − 0.934·9-s − 0.588·11-s + 1.79·13-s + 0.101·15-s − 0.830·17-s + 0.243·21-s − 1.70·23-s − 0.841·25-s + 0.494·27-s − 0.850·29-s − 1.57·31-s + 0.150·33-s + 0.379·35-s + 0.983·37-s − 0.457·39-s + 0.544·41-s + 0.952·43-s + 0.372·45-s + 1.53·47-s − 0.0926·49-s + 0.212·51-s − 0.516·53-s + 0.234·55-s + 0.369·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6518202140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6518202140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.442T + 3T^{2} \) |
| 5 | \( 1 + 0.891T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 - 6.45T + 13T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 29 | \( 1 + 4.58T + 29T^{2} \) |
| 31 | \( 1 + 8.79T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 3.48T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 - 2.84T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 + 2.67T + 67T^{2} \) |
| 71 | \( 1 - 0.0564T + 71T^{2} \) |
| 73 | \( 1 + 6.96T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 1.86T + 89T^{2} \) |
| 97 | \( 1 + 7.82T + 97T^{2} \) |
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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
−8.051042574428329998341226700504, −7.53164673898341101085582776571, −6.48793142718816121970686388065, −5.88216059474160919068777583672, −5.62702273974979018631286880411, −4.17521635757271144730308219596, −3.78369453212802894970607509407, −2.88245218033048436725698515389, −1.92259003843849698666013137307, −0.41128281462146123427715405839,
0.41128281462146123427715405839, 1.92259003843849698666013137307, 2.88245218033048436725698515389, 3.78369453212802894970607509407, 4.17521635757271144730308219596, 5.62702273974979018631286880411, 5.88216059474160919068777583672, 6.48793142718816121970686388065, 7.53164673898341101085582776571, 8.051042574428329998341226700504
