| L(s) = 1 | + 2.47·2-s − 3-s + 4.12·4-s + 3.86·5-s − 2.47·6-s + 4.55·7-s + 5.25·8-s + 9-s + 9.57·10-s + 1.38·11-s − 4.12·12-s − 5.68·13-s + 11.2·14-s − 3.86·15-s + 4.74·16-s − 17-s + 2.47·18-s + 5.64·19-s + 15.9·20-s − 4.55·21-s + 3.43·22-s − 9.45·23-s − 5.25·24-s + 9.96·25-s − 14.0·26-s − 27-s + 18.7·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.06·4-s + 1.73·5-s − 1.01·6-s + 1.71·7-s + 1.85·8-s + 0.333·9-s + 3.02·10-s + 0.418·11-s − 1.18·12-s − 1.57·13-s + 3.00·14-s − 0.998·15-s + 1.18·16-s − 0.242·17-s + 0.583·18-s + 1.29·19-s + 3.56·20-s − 0.992·21-s + 0.731·22-s − 1.97·23-s − 1.07·24-s + 1.99·25-s − 2.76·26-s − 0.192·27-s + 3.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.602217553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.602217553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 + 9.45T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 - 5.97T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 0.951T + 41T^{2} \) |
| 43 | \( 1 + 6.02T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 0.624T + 67T^{2} \) |
| 71 | \( 1 - 2.42T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 83 | \( 1 + 2.89T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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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
−8.114546992420641012423306334734, −7.44395319227205751776430306560, −6.52939221652183264181607378708, −5.95075894325788752261581622190, −5.31569098211297854325765351986, −4.83941500350516423379443608061, −4.32025871832691055247464910195, −2.91927873885727356903917789630, −2.02274190217947888200020122819, −1.56954258149878046084104151450,
1.56954258149878046084104151450, 2.02274190217947888200020122819, 2.91927873885727356903917789630, 4.32025871832691055247464910195, 4.83941500350516423379443608061, 5.31569098211297854325765351986, 5.95075894325788752261581622190, 6.52939221652183264181607378708, 7.44395319227205751776430306560, 8.114546992420641012423306334734
