| L(s) = 1 | − 3-s − 1.48·5-s − 7-s + 9-s − 3.48·11-s + 13-s + 1.48·15-s + 3.19·17-s + 1.19·19-s + 21-s + 1.19·23-s − 2.78·25-s − 27-s + 0.803·29-s − 9.37·31-s + 3.48·33-s + 1.48·35-s − 6.17·37-s − 39-s − 2.68·41-s + 8.17·43-s − 1.48·45-s − 11.0·47-s + 49-s − 3.19·51-s + 11.9·53-s + 5.19·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.666·5-s − 0.377·7-s + 0.333·9-s − 1.05·11-s + 0.277·13-s + 0.384·15-s + 0.775·17-s + 0.274·19-s + 0.218·21-s + 0.249·23-s − 0.556·25-s − 0.192·27-s + 0.149·29-s − 1.68·31-s + 0.607·33-s + 0.251·35-s − 1.01·37-s − 0.160·39-s − 0.419·41-s + 1.24·43-s − 0.222·45-s − 1.61·47-s + 0.142·49-s − 0.447·51-s + 1.64·53-s + 0.700·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8315473789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8315473789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 1.48T + 5T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 1.19T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 - 0.803T + 29T^{2} \) |
| 31 | \( 1 + 9.37T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 43 | \( 1 - 8.17T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 6.97T + 79T^{2} \) |
| 83 | \( 1 - 9.27T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 - 7.95T + 97T^{2} \) |
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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
−8.330099921086061322075035228384, −7.34075453117270795145885525984, −7.25182007500230235877522187200, −5.96477404711793153166636651756, −5.52891608656189291889034261006, −4.70013086500999531717459225717, −3.74648458792021414212091351110, −3.10737454842967697615358025749, −1.86711270327919933286034614561, −0.51626680459590020128201517483,
0.51626680459590020128201517483, 1.86711270327919933286034614561, 3.10737454842967697615358025749, 3.74648458792021414212091351110, 4.70013086500999531717459225717, 5.52891608656189291889034261006, 5.96477404711793153166636651756, 7.25182007500230235877522187200, 7.34075453117270795145885525984, 8.330099921086061322075035228384
