| L(s) = 1 | − 0.518·3-s − 4.15·5-s + 3.03·7-s − 2.73·9-s + 4.16·11-s − 6.95·13-s + 2.15·15-s + 5.57·17-s − 2.60·19-s − 1.57·21-s − 2.78·23-s + 12.2·25-s + 2.97·27-s − 7.68·29-s + 7.14·31-s − 2.15·33-s − 12.6·35-s + 4.89·37-s + 3.61·39-s + 6.93·41-s − 3.39·43-s + 11.3·45-s − 6.56·47-s + 2.22·49-s − 2.89·51-s − 9.31·53-s − 17.3·55-s + ⋯ |
| L(s) = 1 | − 0.299·3-s − 1.86·5-s + 1.14·7-s − 0.910·9-s + 1.25·11-s − 1.93·13-s + 0.557·15-s + 1.35·17-s − 0.596·19-s − 0.343·21-s − 0.580·23-s + 2.45·25-s + 0.572·27-s − 1.42·29-s + 1.28·31-s − 0.375·33-s − 2.13·35-s + 0.804·37-s + 0.578·39-s + 1.08·41-s − 0.517·43-s + 1.69·45-s − 0.958·47-s + 0.318·49-s − 0.404·51-s − 1.27·53-s − 2.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 + 0.518T + 3T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 + 6.95T + 13T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 - 3.48T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 + 7.53T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.65307356591516736409937542177, −6.89219930746023985104254104140, −6.06660093174059578643595095860, −5.05393909215280959902510721117, −4.68334995825071699584984399863, −3.93410652984645526030209827015, −3.23192078456143671779457458718, −2.23791847086260020445116907131, −0.991981930280260707960643527603, 0,
0.991981930280260707960643527603, 2.23791847086260020445116907131, 3.23192078456143671779457458718, 3.93410652984645526030209827015, 4.68334995825071699584984399863, 5.05393909215280959902510721117, 6.06660093174059578643595095860, 6.89219930746023985104254104140, 7.65307356591516736409937542177
