L(s) = 1 | − 25·5-s + 94.1·7-s − 143.·11-s + 421.·13-s − 1.98e3·17-s − 1.31e3·19-s + 4.02e3·23-s + 625·25-s + 6.41e3·29-s − 2.35e3·31-s − 2.35e3·35-s − 7.87e3·37-s − 1.50e4·41-s + 1.14e3·43-s + 2.15e4·47-s − 7.94e3·49-s − 9.56e3·53-s + 3.59e3·55-s − 4.27e4·59-s + 3.21e4·61-s − 1.05e4·65-s − 3.03e4·67-s − 3.60e4·71-s − 6.34e4·73-s − 1.35e4·77-s − 8.99e4·79-s − 3.82e4·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.726·7-s − 0.358·11-s + 0.691·13-s − 1.66·17-s − 0.837·19-s + 1.58·23-s + 0.200·25-s + 1.41·29-s − 0.439·31-s − 0.324·35-s − 0.945·37-s − 1.40·41-s + 0.0941·43-s + 1.42·47-s − 0.472·49-s − 0.467·53-s + 0.160·55-s − 1.59·59-s + 1.10·61-s − 0.309·65-s − 0.826·67-s − 0.847·71-s − 1.39·73-s − 0.260·77-s − 1.62·79-s − 0.608·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 94.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.74e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.78e5T + 8.58e9T^{2} \) |
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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
−10.47615774225553211203635226439, −8.850245100209165565418640370701, −8.527973656946793710134034916434, −7.28293833581298032159706799164, −6.41490095362630303132544092176, −5.03066402310484624616369917382, −4.24183654296505662337599131711, −2.84823631782948368784717679707, −1.49536660072773773982616003823, 0,
1.49536660072773773982616003823, 2.84823631782948368784717679707, 4.24183654296505662337599131711, 5.03066402310484624616369917382, 6.41490095362630303132544092176, 7.28293833581298032159706799164, 8.527973656946793710134034916434, 8.850245100209165565418640370701, 10.47615774225553211203635226439