| 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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.74T + 343T^{2} \) |
| 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
−8.724799315688735591460731773825, −7.82043602654827587803491665629, −7.30092025440511508407231959742, −6.40511679546833741412963600861, −5.08768026533605371090605198207, −4.85289542193259311843993670900, −3.32406267100973751941857449855, −2.75637228871461436459842038280, −1.22956791930666257803616353082, 0,
1.22956791930666257803616353082, 2.75637228871461436459842038280, 3.32406267100973751941857449855, 4.85289542193259311843993670900, 5.08768026533605371090605198207, 6.40511679546833741412963600861, 7.30092025440511508407231959742, 7.82043602654827587803491665629, 8.724799315688735591460731773825
