| L(s) = 1 | + 2.08·2-s + 3-s + 2.33·4-s + 3.00·5-s + 2.08·6-s − 7-s + 0.699·8-s + 9-s + 6.26·10-s − 1.53·11-s + 2.33·12-s + 1.44·13-s − 2.08·14-s + 3.00·15-s − 3.21·16-s − 1.92·17-s + 2.08·18-s + 6.21·19-s + 7.02·20-s − 21-s − 3.19·22-s + 7.68·23-s + 0.699·24-s + 4.05·25-s + 3.01·26-s + 27-s − 2.33·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.16·4-s + 1.34·5-s + 0.850·6-s − 0.377·7-s + 0.247·8-s + 0.333·9-s + 1.98·10-s − 0.462·11-s + 0.674·12-s + 0.401·13-s − 0.556·14-s + 0.776·15-s − 0.803·16-s − 0.465·17-s + 0.490·18-s + 1.42·19-s + 1.57·20-s − 0.218·21-s − 0.680·22-s + 1.60·23-s + 0.142·24-s + 0.810·25-s + 0.590·26-s + 0.192·27-s − 0.441·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.721642213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.721642213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 3.00T + 5T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 - 6.21T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 9.93T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 4.98T + 59T^{2} \) |
| 61 | \( 1 - 0.290T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 8.64T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 5.84T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.651445591114098581998059887304, −7.36287384826909662651159451879, −6.77650432095820639571051660900, −6.01018722273907055228673370420, −5.38745835547646162833842981040, −4.80868070619420738214358907731, −3.80317018701596833374740268944, −2.90433354930020158165516506521, −2.52832624785715259066865891491, −1.29069934536180236309043128071,
1.29069934536180236309043128071, 2.52832624785715259066865891491, 2.90433354930020158165516506521, 3.80317018701596833374740268944, 4.80868070619420738214358907731, 5.38745835547646162833842981040, 6.01018722273907055228673370420, 6.77650432095820639571051660900, 7.36287384826909662651159451879, 8.651445591114098581998059887304
