L(s) = 1 | + (−5.12 − 5.12i)2-s + (−10.3 + 10.3i)3-s + 36.5i·4-s + 106.·6-s + (−13.0 − 13.0i)7-s + (105. − 105. i)8-s − 132. i·9-s − 135.·11-s + (−378. − 378. i)12-s + (35.4 − 35.4i)13-s + 134. i·14-s − 497.·16-s + (−146. − 146. i)17-s + (−680. + 680. i)18-s − 408. i·19-s + ⋯ |
L(s) = 1 | + (−1.28 − 1.28i)2-s + (−1.14 + 1.14i)3-s + 2.28i·4-s + 2.94·6-s + (−0.267 − 0.267i)7-s + (1.64 − 1.64i)8-s − 1.63i·9-s − 1.12·11-s + (−2.62 − 2.62i)12-s + (0.209 − 0.209i)13-s + 0.685i·14-s − 1.94·16-s + (−0.506 − 0.506i)17-s + (−2.10 + 2.10i)18-s − 1.13i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2860051812\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2860051812\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (13.0 + 13.0i)T \) |
good | 2 | \( 1 + (5.12 + 5.12i)T + 16iT^{2} \) |
| 3 | \( 1 + (10.3 - 10.3i)T - 81iT^{2} \) |
| 11 | \( 1 + 135.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-35.4 + 35.4i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (146. + 146. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 408. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (212. - 212. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 195. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 965.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (925. + 925. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 2.29e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.19e3 + 1.19e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-1.55e3 - 1.55e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (645. - 645. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 2.29e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.19e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-6.11e3 - 6.11e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 740.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (1.76e3 - 1.76e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 8.40e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (5.89e3 - 5.89e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 4.76e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (687. + 687. i)T + 8.85e7iT^{2} \) |
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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
−11.52272395501365503065352953584, −10.88710473867831711832449736634, −10.24055589837021082505566213197, −9.530220535123677563183788288660, −8.473998768537776781256882945751, −7.06458201760058168819511782808, −5.34270949518639009863193133118, −4.06712453495741272742248544423, −2.70543492156349084780317969471, −0.60816961254894766566468217977,
0.35951293856581048817887109148, 1.79461738554919540284805983793, 5.18233503787510218057695002952, 6.10551660736810863710365327891, 6.71395721569180632564138553660, 7.81045659243645554190718638359, 8.462919137712394896541382158386, 9.983796849351963734688241009591, 10.72028428392456752505325818929, 11.91994874226571365931741962098