| L(s) = 1 | − 2i·5-s + 4.53·7-s − 4.53·11-s + 4.53i·13-s + 6.53i·17-s − 6.53i·19-s − 0.531i·23-s + 25-s + 2i·29-s − 2i·31-s − 9.06i·35-s + (−5.53 + 2.53i)37-s + 10·41-s + 11.0i·43-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s + 1.71·7-s − 1.36·11-s + 1.25i·13-s + 1.58i·17-s − 1.49i·19-s − 0.110i·23-s + 0.200·25-s + 0.371i·29-s − 0.359i·31-s − 1.53i·35-s + (−0.909 + 0.416i)37-s + 1.56·41-s + 1.68i·43-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142373683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142373683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (5.53 - 2.53i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 4.53T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 - 4.53iT - 13T^{2} \) |
| 17 | \( 1 - 6.53iT - 17T^{2} \) |
| 19 | \( 1 + 6.53iT - 19T^{2} \) |
| 23 | \( 1 + 0.531iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 - 9.06iT - 59T^{2} \) |
| 61 | \( 1 - 5.06iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.06T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 - 5.46iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.223631336448696207226592204452, −7.74036152454324044879779537386, −6.92273961640864361729495674290, −5.91137879129758024744140970275, −5.13266523454300192886200084013, −4.63000378327889383243618218164, −4.13820969212436645943926215366, −2.65872027608481462971139247712, −1.85987561480603163188219160330, −1.02332963524024102730329397663,
0.64283981777209226499621522981, 2.00197938497909073575578836591, 2.68538047236463973380660955810, 3.51964153281441049276340501835, 4.60477504959150222385513419188, 5.45724527925709373725319336029, 5.56885435124281078415895926522, 6.97570668903430373698526186757, 7.53880592348867176467243600602, 8.004503892800768132407098279171
