| L(s) = 1 | + 3-s + 2.54·5-s − 4.05·7-s + 9-s + 0.833·11-s + 2.46·13-s + 2.54·15-s − 1.83·17-s − 5.92·19-s − 4.05·21-s − 3.92·23-s + 1.46·25-s + 27-s − 7.78·29-s − 8.46·31-s + 0.833·33-s − 10.3·35-s + 3.96·37-s + 2.46·39-s − 11.0·41-s − 10.1·43-s + 2.54·45-s + 0.498·47-s + 9.40·49-s − 1.83·51-s + 2.45·53-s + 2.12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·5-s − 1.53·7-s + 0.333·9-s + 0.251·11-s + 0.682·13-s + 0.656·15-s − 0.446·17-s − 1.35·19-s − 0.883·21-s − 0.817·23-s + 0.293·25-s + 0.192·27-s − 1.44·29-s − 1.51·31-s + 0.145·33-s − 1.74·35-s + 0.651·37-s + 0.394·39-s − 1.72·41-s − 1.54·43-s + 0.379·45-s + 0.0726·47-s + 1.34·49-s − 0.257·51-s + 0.337·53-s + 0.285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 2.54T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 - 0.833T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 5.92T + 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 - 3.96T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 0.498T + 47T^{2} \) |
| 53 | \( 1 - 2.45T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 3.97T + 71T^{2} \) |
| 73 | \( 1 - 6.83T + 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 + 4.66T + 83T^{2} \) |
| 89 | \( 1 + 4.56T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.333390893076308603186033526008, −7.15610823911444660490330537122, −6.51503020773670896102596896349, −6.06094850303353407652506154091, −5.25095186554722852427379767092, −3.89823007376272264226664012646, −3.52790058170199856889482901447, −2.33776850107007713865957999060, −1.75956024153102106506725367570, 0,
1.75956024153102106506725367570, 2.33776850107007713865957999060, 3.52790058170199856889482901447, 3.89823007376272264226664012646, 5.25095186554722852427379767092, 6.06094850303353407652506154091, 6.51503020773670896102596896349, 7.15610823911444660490330537122, 8.333390893076308603186033526008
