| L(s) = 1 | − 2.30·3-s + 1.30·5-s + 7-s + 2.30·9-s + 4·11-s + 4.60·13-s − 3·15-s − 17-s + 8.60·19-s − 2.30·21-s − 4·23-s − 3.30·25-s + 1.60·27-s − 9.21·29-s − 7.30·31-s − 9.21·33-s + 1.30·35-s − 9.81·37-s − 10.6·39-s − 11.5·41-s − 4.30·43-s + 3.00·45-s + 2.60·47-s + 49-s + 2.30·51-s − 0.697·53-s + 5.21·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.582·5-s + 0.377·7-s + 0.767·9-s + 1.20·11-s + 1.27·13-s − 0.774·15-s − 0.242·17-s + 1.97·19-s − 0.502·21-s − 0.834·23-s − 0.660·25-s + 0.308·27-s − 1.71·29-s − 1.31·31-s − 1.60·33-s + 0.220·35-s − 1.61·37-s − 1.69·39-s − 1.79·41-s − 0.656·43-s + 0.447·45-s + 0.380·47-s + 0.142·49-s + 0.322·51-s − 0.0957·53-s + 0.702·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 19 | \( 1 - 8.60T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 0.697T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 - 2.60T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.33199144209078969770772627247, −6.64780800915932642799253418033, −6.04202446226652602982199053568, −5.48460072577955926104963849558, −5.04389874402273920405973205053, −3.85814844390508557367636300042, −3.43872623784709645761890811673, −1.72154460828011868006840892944, −1.40086428868647192079999411453, 0,
1.40086428868647192079999411453, 1.72154460828011868006840892944, 3.43872623784709645761890811673, 3.85814844390508557367636300042, 5.04389874402273920405973205053, 5.48460072577955926104963849558, 6.04202446226652602982199053568, 6.64780800915932642799253418033, 7.33199144209078969770772627247
