| L(s) = 1 | − 5·5-s − 24.7·7-s + 8.16·11-s − 46.7·13-s + 60.3·17-s + 111.·19-s − 36.9·23-s + 25·25-s + 33.2·29-s + 124.·31-s + 123.·35-s − 438.·37-s + 508.·41-s + 48.5·43-s + 248.·47-s + 271.·49-s − 320.·53-s − 40.8·55-s + 652.·59-s + 693.·61-s + 233.·65-s + 12.0·67-s − 1.16e3·71-s − 122.·73-s − 202.·77-s + 441.·79-s − 428.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.33·7-s + 0.223·11-s − 0.997·13-s + 0.860·17-s + 1.34·19-s − 0.334·23-s + 0.200·25-s + 0.212·29-s + 0.720·31-s + 0.598·35-s − 1.94·37-s + 1.93·41-s + 0.172·43-s + 0.769·47-s + 0.791·49-s − 0.830·53-s − 0.100·55-s + 1.43·59-s + 1.45·61-s + 0.446·65-s + 0.0219·67-s − 1.95·71-s − 0.195·73-s − 0.299·77-s + 0.629·79-s − 0.566·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 24.7T + 343T^{2} \) |
| 11 | \( 1 - 8.16T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 36.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 438.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 48.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 248.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 320.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 12.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 441.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 428.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 500.T + 9.12e5T^{2} \) |
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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
−8.310770934539453129575664626826, −7.37374695200219440911758462995, −6.97213289207322803188708023747, −5.94027397320684799618793145298, −5.22753459566887183435608429972, −4.11855572336999968744912447344, −3.30107397215558925859693112215, −2.59306058190371649028922223128, −1.04842071290843893553575578125, 0,
1.04842071290843893553575578125, 2.59306058190371649028922223128, 3.30107397215558925859693112215, 4.11855572336999968744912447344, 5.22753459566887183435608429972, 5.94027397320684799618793145298, 6.97213289207322803188708023747, 7.37374695200219440911758462995, 8.310770934539453129575664626826
