| L(s) = 1 | − 5-s − 4.60·7-s − 11-s − 0.605·13-s + 2.60·17-s − 2·19-s + 25-s + 5.21·29-s + 9.21·31-s + 4.60·35-s + 7.21·37-s − 5.21·41-s − 4.60·43-s + 5.21·47-s + 14.2·49-s + 6·53-s + 55-s − 3.21·61-s + 0.605·65-s + 14.4·67-s − 12·71-s + 4.60·73-s + 4.60·77-s − 14·79-s − 9.39·83-s − 2.60·85-s − 16.4·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.74·7-s − 0.301·11-s − 0.167·13-s + 0.631·17-s − 0.458·19-s + 0.200·25-s + 0.967·29-s + 1.65·31-s + 0.778·35-s + 1.18·37-s − 0.813·41-s − 0.702·43-s + 0.760·47-s + 2.03·49-s + 0.824·53-s + 0.134·55-s − 0.411·61-s + 0.0751·65-s + 1.76·67-s − 1.42·71-s + 0.539·73-s + 0.524·77-s − 1.57·79-s − 1.03·83-s − 0.282·85-s − 1.74·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 4.60T + 7T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 + 4.60T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 9.39T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.39316010566265295861166509601, −6.75982877247418833198041529366, −6.23425062675689203714115654617, −5.52218478960576133596550647706, −4.54659573847595067111368138853, −3.86910579489095356437486703493, −3.00118151932031925998973631019, −2.59179614535535003165811060940, −1.02680469340857162915502034470, 0,
1.02680469340857162915502034470, 2.59179614535535003165811060940, 3.00118151932031925998973631019, 3.86910579489095356437486703493, 4.54659573847595067111368138853, 5.52218478960576133596550647706, 6.23425062675689203714115654617, 6.75982877247418833198041529366, 7.39316010566265295861166509601
