| L(s) = 1 | + 15.8·5-s − 15.8·11-s − 26·13-s − 79.3·17-s − 68·19-s − 47.6·23-s + 127.·25-s + 253.·29-s − 212·31-s + 218·37-s + 396.·41-s + 260·43-s + 412.·47-s + 476.·53-s − 252.·55-s − 285.·59-s + 322·61-s − 412.·65-s + 356·67-s + 1.12e3·71-s + 226·73-s + 440·79-s − 253.·83-s − 1.26e3·85-s + 206.·89-s − 1.07e3·95-s + 1.33e3·97-s + ⋯ |
| L(s) = 1 | + 1.41·5-s − 0.435·11-s − 0.554·13-s − 1.13·17-s − 0.821·19-s − 0.431·23-s + 1.01·25-s + 1.62·29-s − 1.22·31-s + 0.968·37-s + 1.51·41-s + 0.922·43-s + 1.28·47-s + 1.23·53-s − 0.617·55-s − 0.630·59-s + 0.675·61-s − 0.787·65-s + 0.649·67-s + 1.88·71-s + 0.362·73-s + 0.626·79-s − 0.335·83-s − 1.60·85-s + 0.245·89-s − 1.16·95-s + 1.39·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.589991993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589991993\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 15.8T + 125T^{2} \) |
| 11 | \( 1 + 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68T + 6.85e3T^{2} \) |
| 23 | \( 1 + 47.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 253.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212T + 2.97e4T^{2} \) |
| 37 | \( 1 - 218T + 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260T + 7.95e4T^{2} \) |
| 47 | \( 1 - 412.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 476.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 322T + 2.26e5T^{2} \) |
| 67 | \( 1 - 356T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 226T + 3.89e5T^{2} \) |
| 79 | \( 1 - 440T + 4.93e5T^{2} \) |
| 83 | \( 1 + 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−9.121133241477848039146635896839, −8.248813656672527165532406525657, −7.25697985233830755910318280207, −6.39807778125663354264012064540, −5.80698638779131516304585397187, −4.92544636211797972646425303001, −4.05599713452408346892322793738, −2.46540358907762550955194191817, −2.21691759668632263475300582672, −0.74287512877540534448815339500,
0.74287512877540534448815339500, 2.21691759668632263475300582672, 2.46540358907762550955194191817, 4.05599713452408346892322793738, 4.92544636211797972646425303001, 5.80698638779131516304585397187, 6.39807778125663354264012064540, 7.25697985233830755910318280207, 8.248813656672527165532406525657, 9.121133241477848039146635896839
