| L(s) = 1 | − 3.43·2-s + 9.14·3-s + 3.79·4-s − 5.89·5-s − 31.4·6-s + 28.3·7-s + 14.4·8-s + 56.6·9-s + 20.2·10-s + 19.6·11-s + 34.6·12-s + 9.96·13-s − 97.2·14-s − 53.8·15-s − 79.9·16-s + 55.5·17-s − 194.·18-s + 137.·19-s − 22.3·20-s + 259.·21-s − 67.5·22-s + 115.·23-s + 132.·24-s − 90.2·25-s − 34.2·26-s + 270.·27-s + 107.·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 1.75·3-s + 0.473·4-s − 0.527·5-s − 2.13·6-s + 1.52·7-s + 0.638·8-s + 2.09·9-s + 0.639·10-s + 0.539·11-s + 0.834·12-s + 0.212·13-s − 1.85·14-s − 0.927·15-s − 1.24·16-s + 0.792·17-s − 2.54·18-s + 1.66·19-s − 0.249·20-s + 2.69·21-s − 0.654·22-s + 1.04·23-s + 1.12·24-s − 0.722·25-s − 0.258·26-s + 1.93·27-s + 0.724·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.628728195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.628728195\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 + 3.43T + 8T^{2} \) |
| 3 | \( 1 - 9.14T + 27T^{2} \) |
| 5 | \( 1 + 5.89T + 125T^{2} \) |
| 7 | \( 1 - 28.3T + 343T^{2} \) |
| 11 | \( 1 - 19.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.96T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 11.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 58.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 355.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 422.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 308.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 372.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 592.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 741.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 850.T + 9.12e5T^{2} \) |
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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
−9.445008669307216081130093933534, −8.887675585387180290135857202243, −8.210097720097758549501586816373, −7.60521488066356056164845622973, −7.23528680136210860707242573534, −5.16385542387535528399048354399, −4.13968094879644060848928916301, −3.19146287290364701346823251029, −1.78098607425827244035244257476, −1.14954847839510008353227017344,
1.14954847839510008353227017344, 1.78098607425827244035244257476, 3.19146287290364701346823251029, 4.13968094879644060848928916301, 5.16385542387535528399048354399, 7.23528680136210860707242573534, 7.60521488066356056164845622973, 8.210097720097758549501586816373, 8.887675585387180290135857202243, 9.445008669307216081130093933534
