| L(s) = 1 | − 0.180·2-s − 1.82·3-s − 1.96·4-s − 2.68·5-s + 0.330·6-s + 0.717·8-s + 0.334·9-s + 0.485·10-s − 2.69·11-s + 3.59·12-s + 4.90·15-s + 3.80·16-s + 4.76·17-s − 0.0604·18-s − 0.188·19-s + 5.28·20-s + 0.487·22-s + 4.39·23-s − 1.30·24-s + 2.21·25-s + 4.86·27-s + 7.08·29-s − 0.887·30-s + 3.69·31-s − 2.12·32-s + 4.91·33-s − 0.861·34-s + ⋯ |
| L(s) = 1 | − 0.127·2-s − 1.05·3-s − 0.983·4-s − 1.20·5-s + 0.134·6-s + 0.253·8-s + 0.111·9-s + 0.153·10-s − 0.812·11-s + 1.03·12-s + 1.26·15-s + 0.951·16-s + 1.15·17-s − 0.0142·18-s − 0.0432·19-s + 1.18·20-s + 0.103·22-s + 0.917·23-s − 0.267·24-s + 0.443·25-s + 0.936·27-s + 1.31·29-s − 0.161·30-s + 0.664·31-s − 0.375·32-s + 0.856·33-s − 0.147·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3387462925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3387462925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.180T + 2T^{2} \) |
| 3 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 + 2.68T + 5T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 + 0.188T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 61 | \( 1 + 0.411T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 - 0.451T + 97T^{2} \) |
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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
−8.034447467612236789404052839347, −7.15329696347352498831751914994, −6.41538798856379692381816640903, −5.49884558745809403468255347254, −4.98238568266355389695256548536, −4.55365210382172638964024375393, −3.50650203132451087035798192219, −3.00043797380912957165832739756, −1.25888486424146368621801481685, −0.35423015336388461633855987576,
0.35423015336388461633855987576, 1.25888486424146368621801481685, 3.00043797380912957165832739756, 3.50650203132451087035798192219, 4.55365210382172638964024375393, 4.98238568266355389695256548536, 5.49884558745809403468255347254, 6.41538798856379692381816640903, 7.15329696347352498831751914994, 8.034447467612236789404052839347
