| L(s) = 1 | − 25·5-s − 74.6·7-s − 25.3·11-s − 448.·13-s + 1.22e3·17-s + 1.17e3·19-s + 677.·23-s + 625·25-s + 1.12e3·29-s + 926.·31-s + 1.86e3·35-s + 5.26e3·37-s + 1.12e3·41-s + 7.38e3·43-s − 1.08e4·47-s − 1.12e4·49-s − 2.38e4·53-s + 632.·55-s + 2.07e4·59-s − 2.80e4·61-s + 1.12e4·65-s + 6.26e4·67-s − 2.04e4·71-s − 2.01e4·73-s + 1.88e3·77-s + 1.05e4·79-s − 3.88e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.576·7-s − 0.0630·11-s − 0.735·13-s + 1.02·17-s + 0.744·19-s + 0.267·23-s + 0.200·25-s + 0.249·29-s + 0.173·31-s + 0.257·35-s + 0.632·37-s + 0.104·41-s + 0.609·43-s − 0.715·47-s − 0.667·49-s − 1.16·53-s + 0.0281·55-s + 0.776·59-s − 0.966·61-s + 0.328·65-s + 1.70·67-s − 0.481·71-s − 0.442·73-s + 0.0363·77-s + 0.189·79-s − 0.618·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 25T \) |
| good | 7 | \( 1 + 74.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 25.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 448.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 677.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 926.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.38e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.05e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.18e4T + 8.58e9T^{2} \) |
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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
−9.435314214932669761072095408067, −8.220024392666879097853785131793, −7.52525834719827687486481116548, −6.65683177161751244756422688320, −5.60165266599212474151595758701, −4.67010477793506207902737940525, −3.51091873538006895972686011476, −2.69520259932036438925974912042, −1.16844679893818808394219240793, 0,
1.16844679893818808394219240793, 2.69520259932036438925974912042, 3.51091873538006895972686011476, 4.67010477793506207902737940525, 5.60165266599212474151595758701, 6.65683177161751244756422688320, 7.52525834719827687486481116548, 8.220024392666879097853785131793, 9.435314214932669761072095408067
