| L(s) = 1 | + 17.4·5-s + 2.99·7-s − 10.6·11-s + 43.3·13-s + 37.8·17-s − 79.8·19-s − 191.·23-s + 178.·25-s + 138.·29-s + 212.·31-s + 52.1·35-s + 270.·37-s − 441.·41-s − 64.1·43-s + 436.·47-s − 334.·49-s + 278.·53-s − 185.·55-s + 830.·59-s + 724.·61-s + 754.·65-s − 859.·67-s + 681.·71-s + 785.·73-s − 31.7·77-s − 1.01e3·79-s + 467.·83-s + ⋯ |
| L(s) = 1 | + 1.55·5-s + 0.161·7-s − 0.291·11-s + 0.924·13-s + 0.540·17-s − 0.964·19-s − 1.73·23-s + 1.43·25-s + 0.889·29-s + 1.22·31-s + 0.251·35-s + 1.20·37-s − 1.68·41-s − 0.227·43-s + 1.35·47-s − 0.973·49-s + 0.721·53-s − 0.454·55-s + 1.83·59-s + 1.52·61-s + 1.44·65-s − 1.56·67-s + 1.13·71-s + 1.25·73-s − 0.0470·77-s − 1.45·79-s + 0.618·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.427562309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.427562309\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 17.4T + 125T^{2} \) |
| 7 | \( 1 - 2.99T + 343T^{2} \) |
| 11 | \( 1 + 10.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 278.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 830.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 724.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 859.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 467.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 510.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 234.T + 9.12e5T^{2} \) |
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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
−8.468241983073840029239391487176, −8.228864676621715530525422622192, −6.87574618456272154076281832665, −6.16741131270362332517816639443, −5.72230119698521909991970881022, −4.77180111777568215562226885720, −3.78399194275148997442595816730, −2.56973628419862132670841545741, −1.88507311928098840215928641225, −0.848926869569398604333360920792,
0.848926869569398604333360920792, 1.88507311928098840215928641225, 2.56973628419862132670841545741, 3.78399194275148997442595816730, 4.77180111777568215562226885720, 5.72230119698521909991970881022, 6.16741131270362332517816639443, 6.87574618456272154076281832665, 8.228864676621715530525422622192, 8.468241983073840029239391487176
