| L(s) = 1 | + (−0.236 + 0.219i)2-s + (7.21 − 1.08i)3-s + (−0.590 + 7.87i)4-s + (−5.66 + 14.4i)5-s + (−1.46 + 1.83i)6-s + (5.75 − 17.6i)7-s + (−3.19 − 4.00i)8-s + (25.0 − 7.72i)9-s + (−1.82 − 4.64i)10-s + (24.5 + 7.57i)11-s + (4.30 + 57.4i)12-s + (−8.29 − 36.3i)13-s + (2.49 + 5.41i)14-s + (−25.1 + 110. i)15-s + (−60.8 − 9.16i)16-s + (85.3 − 58.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0834 + 0.0774i)2-s + (1.38 − 0.209i)3-s + (−0.0737 + 0.984i)4-s + (−0.506 + 1.29i)5-s + (−0.0996 + 0.124i)6-s + (0.310 − 0.950i)7-s + (−0.141 − 0.176i)8-s + (0.927 − 0.286i)9-s + (−0.0576 − 0.147i)10-s + (0.673 + 0.207i)11-s + (0.103 + 1.38i)12-s + (−0.176 − 0.775i)13-s + (0.0476 + 0.103i)14-s + (−0.433 + 1.89i)15-s + (−0.950 − 0.143i)16-s + (1.21 − 0.830i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.66465 + 0.662648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66465 + 0.662648i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.75 + 17.6i)T \) |
| good | 2 | \( 1 + (0.236 - 0.219i)T + (0.597 - 7.97i)T^{2} \) |
| 3 | \( 1 + (-7.21 + 1.08i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (5.66 - 14.4i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (-24.5 - 7.57i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (8.29 + 36.3i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-85.3 + 58.2i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (24.5 + 42.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (26.6 + 18.1i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (83.0 + 39.9i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (153. - 266. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (16.0 + 213. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (-139. - 174. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (22.1 - 27.8i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-363. + 337. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (12.4 - 165. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-303. - 773. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-39.6 - 529. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (-250. + 433. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (946. - 456. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (450. + 417. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (638. + 1.10e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-6.87 + 30.1i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-654. + 201. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 640.T + 9.12e5T^{2} \) |
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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
−14.84693054145741292758581530226, −14.28941449426042828989877582746, −13.23319851203328223590608233521, −11.82924020754860396662724549774, −10.44429279911161394147556831078, −8.913618544611904670664364344986, −7.50502873218686230632506397469, −7.29251415576922114915707792587, −3.83184000831027882268740545144, −2.90361612090558437027817805346,
1.74442649845016798806011608843, 4.09201596879899616191414083556, 5.68251580623187820449734145197, 8.079775112381186955011362000008, 8.953540495043442479291335045290, 9.650583871872593368153991277991, 11.57126676665661358699979978419, 12.74941555901571279201566857916, 14.22838439917916543367046784685, 14.78476863283329644210845652258
