| L(s) = 1 | + 3·3-s − 11.4·5-s − 13.4·7-s + 9·9-s − 10.9·11-s + 2·13-s − 34.4·15-s + 106.·17-s + 86.9·19-s − 40.4·21-s + 64.9·23-s + 7.06·25-s + 27·27-s + 129.·29-s + 246.·31-s − 32.9·33-s + 155.·35-s − 259.·37-s + 6·39-s + 324.·41-s + 292.·43-s − 103.·45-s + 386.·47-s − 160.·49-s + 320.·51-s − 536.·53-s + 126.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.02·5-s − 0.728·7-s + 0.333·9-s − 0.301·11-s + 0.0426·13-s − 0.593·15-s + 1.52·17-s + 1.05·19-s − 0.420·21-s + 0.588·23-s + 0.0565·25-s + 0.192·27-s + 0.828·29-s + 1.42·31-s − 0.173·33-s + 0.748·35-s − 1.15·37-s + 0.0246·39-s + 1.23·41-s + 1.03·43-s − 0.342·45-s + 1.20·47-s − 0.469·49-s + 0.880·51-s − 1.39·53-s + 0.309·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.785565952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785565952\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 11.4T + 125T^{2} \) |
| 7 | \( 1 + 13.4T + 343T^{2} \) |
| 11 | \( 1 + 10.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 64.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 292.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 628.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 719.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 200.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 987.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 844.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3T + 9.12e5T^{2} \) |
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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
−10.85174077876742916848302143955, −9.917841670146956307383134738932, −9.098237043513211764441474325898, −7.914173847950976842863271434784, −7.49366070289577557140981913858, −6.21811533540754762992433759763, −4.88131481809258899617374581516, −3.61325940369268083479255339715, −2.89995981997098525855231045958, −0.873142756858278007599259612752,
0.873142756858278007599259612752, 2.89995981997098525855231045958, 3.61325940369268083479255339715, 4.88131481809258899617374581516, 6.21811533540754762992433759763, 7.49366070289577557140981913858, 7.914173847950976842863271434784, 9.098237043513211764441474325898, 9.917841670146956307383134738932, 10.85174077876742916848302143955
