| L(s) = 1 | + (−0.593 + 0.804i)2-s + (2.98 − 0.564i)3-s + (−0.294 − 0.955i)4-s + (1.30 + 1.81i)5-s + (−1.31 + 2.73i)6-s + (−1.78 + 1.95i)7-s + (0.943 + 0.330i)8-s + (5.78 − 2.27i)9-s + (−2.23 − 0.0280i)10-s + (−1.15 + 2.94i)11-s + (−1.41 − 2.68i)12-s + (1.56 + 0.176i)13-s + (−0.512 − 2.59i)14-s + (4.91 + 4.68i)15-s + (−0.826 + 0.563i)16-s + (−3.21 − 0.120i)17-s + ⋯ |
| L(s) = 1 | + (−0.419 + 0.568i)2-s + (1.72 − 0.325i)3-s + (−0.147 − 0.477i)4-s + (0.583 + 0.811i)5-s + (−0.537 + 1.11i)6-s + (−0.674 + 0.738i)7-s + (0.333 + 0.116i)8-s + (1.92 − 0.757i)9-s + (−0.707 − 0.00887i)10-s + (−0.348 + 0.888i)11-s + (−0.409 − 0.774i)12-s + (0.433 + 0.0488i)13-s + (−0.137 − 0.693i)14-s + (1.26 + 1.20i)15-s + (−0.206 + 0.140i)16-s + (−0.779 − 0.0291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74283 + 1.03801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74283 + 1.03801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.593 - 0.804i)T \) |
| 5 | \( 1 + (-1.30 - 1.81i)T \) |
| 7 | \( 1 + (1.78 - 1.95i)T \) |
| good | 3 | \( 1 + (-2.98 + 0.564i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (1.15 - 2.94i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 0.176i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (3.21 + 0.120i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-0.602 + 1.04i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.207 + 5.53i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (4.85 - 1.10i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-7.18 + 4.14i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.12 + 4.82i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (2.49 + 5.18i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.70 + 10.5i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-4.33 - 3.19i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-5.11 - 2.70i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.468 - 6.25i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (1.92 - 6.24i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (14.3 + 3.83i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.494 + 2.16i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (5.02 - 3.70i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-4.24 - 2.45i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.286 - 2.54i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-4.91 - 12.5i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (6.85 + 6.85i)T + 97iT^{2} \) |
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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
−10.67448548156341254950229902603, −9.837322692216044996826590138561, −9.176173812911137801553364401661, −8.555169180874036574044672000476, −7.45680105495270756516660476740, −6.83936555109168171605380087052, −5.86929233921112989636425156462, −4.16299165633184702001969008575, −2.73713095460721667173852412693, −2.13188776791166515892470880848,
1.36296217080681147254056482204, 2.77207008046501367693394618914, 3.61726142081912093880378549344, 4.63486696033109393298847008152, 6.28889296345110735236596196796, 7.71361375977687832136816613253, 8.333861830538649730665826881223, 9.141867661186970005517456072827, 9.725324452380441660298672953003, 10.39776059405670355690121243487
