| L(s) = 1 | − 3-s − 4.70·7-s + 9-s − 4.70·11-s − 2.70·13-s − 0.701·17-s + 4.70·19-s + 4.70·21-s − 1.29·23-s − 5·25-s − 27-s + 8·29-s + 9.40·31-s + 4.70·33-s + 37-s + 2.70·39-s + 2·41-s − 4·43-s − 1.40·47-s + 15.1·49-s + 0.701·51-s + 6.70·53-s − 4.70·57-s − 3.40·59-s + 6·61-s − 4.70·63-s + 2.59·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.77·7-s + 0.333·9-s − 1.41·11-s − 0.749·13-s − 0.170·17-s + 1.07·19-s + 1.02·21-s − 0.270·23-s − 25-s − 0.192·27-s + 1.48·29-s + 1.68·31-s + 0.818·33-s + 0.164·37-s + 0.432·39-s + 0.312·41-s − 0.609·43-s − 0.204·47-s + 2.15·49-s + 0.0982·51-s + 0.920·53-s − 0.622·57-s − 0.443·59-s + 0.768·61-s − 0.592·63-s + 0.317·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6999661162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6999661162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 + 0.701T + 17T^{2} \) |
| 19 | \( 1 - 4.70T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 9.40T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−9.722887655888399932317502225141, −8.461829906023158474795962782327, −7.57878596142603729807709001299, −6.82898785232762015612828845703, −6.08269637991764331603323137355, −5.34444473260172080291025242396, −4.41567813981223773172450946413, −3.18490113056377681171550381060, −2.51975725437602115827676653242, −0.55544463708965224718689491760,
0.55544463708965224718689491760, 2.51975725437602115827676653242, 3.18490113056377681171550381060, 4.41567813981223773172450946413, 5.34444473260172080291025242396, 6.08269637991764331603323137355, 6.82898785232762015612828845703, 7.57878596142603729807709001299, 8.461829906023158474795962782327, 9.722887655888399932317502225141
