| L(s) = 1 | + 2.70·2-s + 3-s + 5.34·4-s + 5-s + 2.70·6-s − 1.07·7-s + 9.04·8-s + 9-s + 2.70·10-s + 5.34·12-s + 4.34·13-s − 2.92·14-s + 15-s + 13.8·16-s − 7.75·17-s + 2.70·18-s − 5.26·19-s + 5.34·20-s − 1.07·21-s − 2.15·23-s + 9.04·24-s + 25-s + 11.7·26-s + 27-s − 5.75·28-s − 1.41·29-s + 2.70·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.577·3-s + 2.67·4-s + 0.447·5-s + 1.10·6-s − 0.407·7-s + 3.19·8-s + 0.333·9-s + 0.856·10-s + 1.54·12-s + 1.20·13-s − 0.780·14-s + 0.258·15-s + 3.45·16-s − 1.88·17-s + 0.638·18-s − 1.20·19-s + 1.19·20-s − 0.235·21-s − 0.449·23-s + 1.84·24-s + 0.200·25-s + 2.30·26-s + 0.192·27-s − 1.08·28-s − 0.263·29-s + 0.494·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.328020696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.328020696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 + 7.75T + 17T^{2} \) |
| 19 | \( 1 + 5.26T + 19T^{2} \) |
| 23 | \( 1 + 2.15T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 4.68T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 - 0.156T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.184055150260394680022112465231, −8.417074726670171228732884791717, −7.30969757836230906365548789429, −6.40870015602004296514712804773, −6.18784451316645286782615283200, −5.04916161274286318029216841143, −4.14849204192772368969300782728, −3.61665316948219825277698527930, −2.49282677479250398565505451754, −1.84048687588972411031430750988,
1.84048687588972411031430750988, 2.49282677479250398565505451754, 3.61665316948219825277698527930, 4.14849204192772368969300782728, 5.04916161274286318029216841143, 6.18784451316645286782615283200, 6.40870015602004296514712804773, 7.30969757836230906365548789429, 8.417074726670171228732884791717, 9.184055150260394680022112465231
