| L(s) = 1 | − 0.132·2-s + 3.71·3-s − 7.98·4-s − 2.43·5-s − 0.490·6-s + 30.9·7-s + 2.11·8-s − 13.2·9-s + 0.321·10-s + 57.4·11-s − 29.6·12-s − 27.0·13-s − 4.09·14-s − 9.02·15-s + 63.5·16-s − 91.1·17-s + 1.75·18-s − 77.7·19-s + 19.4·20-s + 114.·21-s − 7.60·22-s + 46.4·23-s + 7.84·24-s − 119.·25-s + 3.57·26-s − 149.·27-s − 246.·28-s + ⋯ |
| L(s) = 1 | − 0.0467·2-s + 0.714·3-s − 0.997·4-s − 0.217·5-s − 0.0333·6-s + 1.66·7-s + 0.0934·8-s − 0.490·9-s + 0.0101·10-s + 1.57·11-s − 0.712·12-s − 0.576·13-s − 0.0781·14-s − 0.155·15-s + 0.993·16-s − 1.30·17-s + 0.0229·18-s − 0.938·19-s + 0.217·20-s + 1.19·21-s − 0.0736·22-s + 0.421·23-s + 0.0667·24-s − 0.952·25-s + 0.0269·26-s − 1.06·27-s − 1.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.132T + 8T^{2} \) |
| 3 | \( 1 - 3.71T + 27T^{2} \) |
| 5 | \( 1 + 2.43T + 125T^{2} \) |
| 7 | \( 1 - 30.9T + 343T^{2} \) |
| 11 | \( 1 - 57.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.94T + 6.89e4T^{2} \) |
| 47 | \( 1 + 485.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 111.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 544.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 367.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 272.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 37.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 667.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 499.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 432.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.30e3T + 9.12e5T^{2} \) |
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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
−8.618485877394462794478959359513, −8.040643994497587960673520970276, −7.11462691072117542457649118090, −6.03332594107210824730946912775, −4.91347637913940669845163695835, −4.37705733816484982842556604846, −3.65091904397105701899754918364, −2.27537234084120989222980366677, −1.39060556511566641332602570905, 0,
1.39060556511566641332602570905, 2.27537234084120989222980366677, 3.65091904397105701899754918364, 4.37705733816484982842556604846, 4.91347637913940669845163695835, 6.03332594107210824730946912775, 7.11462691072117542457649118090, 8.040643994497587960673520970276, 8.618485877394462794478959359513
