| L(s) = 1 | + 5·5-s + 1.74·7-s − 28.9·11-s − 48.2·13-s − 16.2·17-s − 130.·19-s + 182.·23-s + 25·25-s − 291.·29-s + 219.·31-s + 8.73·35-s + 436.·37-s + 339.·41-s + 316.·43-s − 335.·47-s − 339.·49-s + 520.·53-s − 144.·55-s + 589.·59-s − 566.·61-s − 241.·65-s + 407.·67-s + 486.·71-s + 143.·73-s − 50.6·77-s + 968.·79-s − 532.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.0943·7-s − 0.794·11-s − 1.02·13-s − 0.231·17-s − 1.57·19-s + 1.65·23-s + 0.200·25-s − 1.86·29-s + 1.27·31-s + 0.0422·35-s + 1.93·37-s + 1.29·41-s + 1.12·43-s − 1.04·47-s − 0.991·49-s + 1.34·53-s − 0.355·55-s + 1.30·59-s − 1.18·61-s − 0.460·65-s + 0.742·67-s + 0.812·71-s + 0.229·73-s − 0.0749·77-s + 1.37·79-s − 0.704·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.838918477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.838918477\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 1.74T + 343T^{2} \) |
| 11 | \( 1 + 28.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 219.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 339.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 316.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 520.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 566.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 486.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 143.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 968.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 532.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 67.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 218.T + 9.12e5T^{2} \) |
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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
−9.308512139555393232182662598428, −8.327354213251203685812491872786, −7.56927378934543411240532753381, −6.73298807393961043426792631396, −5.84036727135581625349582303971, −4.97389523542831345951528206394, −4.21224768658033662536849521249, −2.78005104036952292404722075943, −2.15220079774395696503825157241, −0.64710079732116236320918425889,
0.64710079732116236320918425889, 2.15220079774395696503825157241, 2.78005104036952292404722075943, 4.21224768658033662536849521249, 4.97389523542831345951528206394, 5.84036727135581625349582303971, 6.73298807393961043426792631396, 7.56927378934543411240532753381, 8.327354213251203685812491872786, 9.308512139555393232182662598428
