| L(s) = 1 | + 3·3-s − 17.4·5-s − 2.99·7-s + 9·9-s − 10.6·11-s + 43.3·13-s − 52.2·15-s − 37.8·17-s + 79.8·19-s − 8.97·21-s − 191.·23-s + 178.·25-s + 27·27-s − 138.·29-s − 212.·31-s − 31.8·33-s + 52.1·35-s + 270.·37-s + 129.·39-s + 441.·41-s + 64.1·43-s − 156.·45-s + 436.·47-s − 334.·49-s − 113.·51-s − 278.·53-s + 185.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.55·5-s − 0.161·7-s + 0.333·9-s − 0.291·11-s + 0.924·13-s − 0.900·15-s − 0.540·17-s + 0.964·19-s − 0.0932·21-s − 1.73·23-s + 1.43·25-s + 0.192·27-s − 0.889·29-s − 1.22·31-s − 0.168·33-s + 0.251·35-s + 1.20·37-s + 0.533·39-s + 1.68·41-s + 0.227·43-s − 0.519·45-s + 1.35·47-s − 0.973·49-s − 0.312·51-s − 0.721·53-s + 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.563666867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.563666867\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + 17.4T + 125T^{2} \) |
| 7 | \( 1 + 2.99T + 343T^{2} \) |
| 11 | \( 1 + 10.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 278.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 830.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 724.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 859.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 467.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 510.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 234.T + 9.12e5T^{2} \) |
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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
−9.790013711174939986606097563544, −8.985320631634665975047992931707, −7.949971113351626837249934008787, −7.73403528080554486348775230803, −6.60826466676024267853482679840, −5.40638440282951443442776051888, −3.94553416345458711921712888889, −3.76330207536323289235158266297, −2.34216783460894381031231013229, −0.67657978050211941420765957109,
0.67657978050211941420765957109, 2.34216783460894381031231013229, 3.76330207536323289235158266297, 3.94553416345458711921712888889, 5.40638440282951443442776051888, 6.60826466676024267853482679840, 7.73403528080554486348775230803, 7.949971113351626837249934008787, 8.985320631634665975047992931707, 9.790013711174939986606097563544
