| L(s) = 1 | − 3.18·3-s + 3.43·5-s + 7-s + 7.16·9-s − 2.57·11-s − 1.58·13-s − 10.9·15-s − 4.77·17-s + 8.65·19-s − 3.18·21-s + 23-s + 6.80·25-s − 13.2·27-s + 7.67·29-s − 5.02·31-s + 8.21·33-s + 3.43·35-s + 3.71·37-s + 5.06·39-s + 11.9·41-s + 7.46·43-s + 24.6·45-s − 4.43·47-s + 49-s + 15.2·51-s + 8.06·53-s − 8.84·55-s + ⋯ |
| L(s) = 1 | − 1.84·3-s + 1.53·5-s + 0.377·7-s + 2.38·9-s − 0.776·11-s − 0.440·13-s − 2.82·15-s − 1.15·17-s + 1.98·19-s − 0.695·21-s + 0.208·23-s + 1.36·25-s − 2.55·27-s + 1.42·29-s − 0.902·31-s + 1.42·33-s + 0.580·35-s + 0.611·37-s + 0.811·39-s + 1.86·41-s + 1.13·43-s + 3.66·45-s − 0.647·47-s + 0.142·49-s + 2.13·51-s + 1.10·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093258008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093258008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 - 8.65T + 19T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 - 3.71T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 2.01T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−10.67572230378846838644441246592, −9.887151237964411799034307357173, −9.250055380600534164036836966144, −7.60125171882261373214136427978, −6.74369143360798783337367867935, −5.83224777025047671949294055186, −5.31771153946630838705608036104, −4.56754226137091210070241163549, −2.43400581438374559371773621780, −1.03917200227646892140178026851,
1.03917200227646892140178026851, 2.43400581438374559371773621780, 4.56754226137091210070241163549, 5.31771153946630838705608036104, 5.83224777025047671949294055186, 6.74369143360798783337367867935, 7.60125171882261373214136427978, 9.250055380600534164036836966144, 9.887151237964411799034307357173, 10.67572230378846838644441246592
