| L(s) = 1 | − 2-s + 3-s + 4-s + 1.20·5-s − 6-s − 1.77·7-s − 8-s + 9-s − 1.20·10-s − 5.30·11-s + 12-s + 13-s + 1.77·14-s + 1.20·15-s + 16-s − 1.91·17-s − 18-s − 1.21·19-s + 1.20·20-s − 1.77·21-s + 5.30·22-s + 5.37·23-s − 24-s − 3.54·25-s − 26-s + 27-s − 1.77·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.539·5-s − 0.408·6-s − 0.669·7-s − 0.353·8-s + 0.333·9-s − 0.381·10-s − 1.60·11-s + 0.288·12-s + 0.277·13-s + 0.473·14-s + 0.311·15-s + 0.250·16-s − 0.464·17-s − 0.235·18-s − 0.277·19-s + 0.269·20-s − 0.386·21-s + 1.13·22-s + 1.12·23-s − 0.204·24-s − 0.708·25-s − 0.196·26-s + 0.192·27-s − 0.334·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325403254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325403254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 + 1.77T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 + 4.77T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.99T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 - 4.86T + 53T^{2} \) |
| 59 | \( 1 - 5.18T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 + 0.631T + 67T^{2} \) |
| 71 | \( 1 - 6.75T + 71T^{2} \) |
| 73 | \( 1 - 4.72T + 73T^{2} \) |
| 79 | \( 1 + 7.71T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 2.67T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.79196444715186931095082983237, −7.35232768116199243874080314300, −6.62172733940378464642224363958, −5.74489102415192151214967859882, −5.29259349318287036444553413748, −4.11348280547352881826687146199, −3.27849014960464467728444939623, −2.46995603917498430743547633412, −1.96508357895394358614209382011, −0.58461260600654690646747844994,
0.58461260600654690646747844994, 1.96508357895394358614209382011, 2.46995603917498430743547633412, 3.27849014960464467728444939623, 4.11348280547352881826687146199, 5.29259349318287036444553413748, 5.74489102415192151214967859882, 6.62172733940378464642224363958, 7.35232768116199243874080314300, 7.79196444715186931095082983237
