| L(s) = 1 | − 3.23·5-s + 0.763·7-s − 4.47·11-s + 13-s + 4.47·17-s − 3.23·19-s − 6.47·23-s + 5.47·25-s + 0.472·29-s + 4.76·31-s − 2.47·35-s − 4.47·37-s + 4.76·41-s + 2.47·43-s − 8.47·47-s − 6.41·49-s − 8.47·53-s + 14.4·55-s + 10.9·59-s − 12.4·61-s − 3.23·65-s − 5.70·67-s − 10·71-s + 4.47·73-s − 3.41·77-s − 8.94·79-s + 14.9·83-s + ⋯ |
| L(s) = 1 | − 1.44·5-s + 0.288·7-s − 1.34·11-s + 0.277·13-s + 1.08·17-s − 0.742·19-s − 1.34·23-s + 1.09·25-s + 0.0876·29-s + 0.855·31-s − 0.417·35-s − 0.735·37-s + 0.744·41-s + 0.376·43-s − 1.23·47-s − 0.916·49-s − 1.16·53-s + 1.95·55-s + 1.42·59-s − 1.59·61-s − 0.401·65-s − 0.697·67-s − 1.18·71-s + 0.523·73-s − 0.389·77-s − 1.00·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8237217121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8237217121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 0.763T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 2.29T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
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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
−7.926489408460554011770534644484, −7.54049714042846935094025442442, −6.51714408628138160579642281118, −5.79470334361140405204788218373, −4.90870218899717124800131522706, −4.35857028531152284536671161288, −3.52965832653256396852235581183, −2.88802539733523814497018709873, −1.77520191729155550262632348087, −0.44470686000457059600434419072,
0.44470686000457059600434419072, 1.77520191729155550262632348087, 2.88802539733523814497018709873, 3.52965832653256396852235581183, 4.35857028531152284536671161288, 4.90870218899717124800131522706, 5.79470334361140405204788218373, 6.51714408628138160579642281118, 7.54049714042846935094025442442, 7.926489408460554011770534644484
