L(s) = 1 | − 2.10·2-s − 1.35·3-s − 3.57·4-s − 5·5-s + 2.84·6-s − 13.9·7-s + 24.3·8-s − 25.1·9-s + 10.5·10-s − 11·11-s + 4.82·12-s + 44.0·13-s + 29.3·14-s + 6.75·15-s − 22.6·16-s − 43.5·17-s + 52.9·18-s + 19·19-s + 17.8·20-s + 18.8·21-s + 23.1·22-s − 78.4·23-s − 32.9·24-s + 25·25-s − 92.7·26-s + 70.5·27-s + 49.8·28-s + ⋯ |
L(s) = 1 | − 0.743·2-s − 0.260·3-s − 0.446·4-s − 0.447·5-s + 0.193·6-s − 0.752·7-s + 1.07·8-s − 0.932·9-s + 0.332·10-s − 0.301·11-s + 0.116·12-s + 0.940·13-s + 0.559·14-s + 0.116·15-s − 0.353·16-s − 0.621·17-s + 0.693·18-s + 0.229·19-s + 0.199·20-s + 0.195·21-s + 0.224·22-s − 0.710·23-s − 0.279·24-s + 0.200·25-s − 0.699·26-s + 0.502·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 2.10T + 8T^{2} \) |
| 3 | \( 1 + 1.35T + 27T^{2} \) |
| 7 | \( 1 + 13.9T + 343T^{2} \) |
| 13 | \( 1 - 44.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 78.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 277.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 293.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 265.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 384.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 472.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 617.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 24.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 828.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 350.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 475.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 757.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 182.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 161.T + 9.12e5T^{2} \) |
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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
−9.096593866770828933225175214225, −8.347789066434480712665029457585, −7.78348640422955987533210337886, −6.57601188507842045496533285751, −5.83989285977296845573851948933, −4.69010053106848791105601472676, −3.77237616558136031655062222686, −2.62765413048744347356794725217, −0.938932031316093563205343103975, 0,
0.938932031316093563205343103975, 2.62765413048744347356794725217, 3.77237616558136031655062222686, 4.69010053106848791105601472676, 5.83989285977296845573851948933, 6.57601188507842045496533285751, 7.78348640422955987533210337886, 8.347789066434480712665029457585, 9.096593866770828933225175214225