| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.049796009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.049796009\) |
| \(L(\frac{3}{2})\) |
|
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
−8.558778459776546118642854288772, −7.81905200637318942706002531313, −6.91180031744610243025595520400, −6.14185501284537459441623711302, −5.70369379927494741227301311202, −4.82003791747613209871908947180, −3.89097246789153798383020306462, −2.88006693389607723093110688272, −1.85545507379180417618131183004, −1.13432136684212882807748148975,
1.13432136684212882807748148975, 1.85545507379180417618131183004, 2.88006693389607723093110688272, 3.89097246789153798383020306462, 4.82003791747613209871908947180, 5.70369379927494741227301311202, 6.14185501284537459441623711302, 6.91180031744610243025595520400, 7.81905200637318942706002531313, 8.558778459776546118642854288772
