| L(s) = 1 | − 0.772·2-s + 8.82·3-s − 7.40·4-s − 17.1·5-s − 6.81·6-s − 3.36·7-s + 11.8·8-s + 50.9·9-s + 13.2·10-s − 65.3·12-s − 13·13-s + 2.59·14-s − 151.·15-s + 50.0·16-s − 109.·17-s − 39.3·18-s + 126.·19-s + 126.·20-s − 29.6·21-s + 37.6·23-s + 105.·24-s + 169.·25-s + 10.0·26-s + 211.·27-s + 24.8·28-s + 208.·29-s + 116.·30-s + ⋯ |
| L(s) = 1 | − 0.273·2-s + 1.69·3-s − 0.925·4-s − 1.53·5-s − 0.463·6-s − 0.181·7-s + 0.525·8-s + 1.88·9-s + 0.418·10-s − 1.57·12-s − 0.277·13-s + 0.0495·14-s − 2.60·15-s + 0.781·16-s − 1.55·17-s − 0.515·18-s + 1.53·19-s + 1.41·20-s − 0.308·21-s + 0.341·23-s + 0.893·24-s + 1.35·25-s + 0.0757·26-s + 1.50·27-s + 0.167·28-s + 1.33·29-s + 0.711·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 0.772T + 8T^{2} \) |
| 3 | \( 1 - 8.82T + 27T^{2} \) |
| 5 | \( 1 + 17.1T + 125T^{2} \) |
| 7 | \( 1 + 3.36T + 343T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 37.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 28.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 12.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 516.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 483.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 798.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 76.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 243.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 91.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 335.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 417.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 345.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 596.T + 9.12e5T^{2} \) |
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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
−8.472257111339081452380199078036, −8.113110794604881937933883117797, −7.51663041237977199455320285184, −6.63614572516431704985191255967, −4.71234332603188799415890219901, −4.42384569168545130189973153992, −3.37583553911625949970759132771, −2.84897098866373611300476571732, −1.25098574316622732982977431931, 0,
1.25098574316622732982977431931, 2.84897098866373611300476571732, 3.37583553911625949970759132771, 4.42384569168545130189973153992, 4.71234332603188799415890219901, 6.63614572516431704985191255967, 7.51663041237977199455320285184, 8.113110794604881937933883117797, 8.472257111339081452380199078036
