| L(s) = 1 | + (−0.0523 − 0.998i)2-s + (−0.158 − 0.128i)3-s + (−0.994 + 0.104i)4-s + (−1.20 + 1.88i)5-s + (−0.119 + 0.164i)6-s + (0.823 − 2.51i)7-s + (0.156 + 0.987i)8-s + (−0.615 − 2.89i)9-s + (1.94 + 1.10i)10-s + (−4.28 − 0.911i)11-s + (0.170 + 0.110i)12-s + (−1.59 − 3.13i)13-s + (−2.55 − 0.690i)14-s + (0.432 − 0.143i)15-s + (0.978 − 0.207i)16-s + (3.12 + 1.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.0370 − 0.706i)2-s + (−0.0913 − 0.0739i)3-s + (−0.497 + 0.0522i)4-s + (−0.539 + 0.842i)5-s + (−0.0488 + 0.0672i)6-s + (0.311 − 0.950i)7-s + (0.0553 + 0.349i)8-s + (−0.205 − 0.964i)9-s + (0.614 + 0.349i)10-s + (−1.29 − 0.274i)11-s + (0.0492 + 0.0320i)12-s + (−0.443 − 0.870i)13-s + (−0.682 − 0.184i)14-s + (0.111 − 0.0370i)15-s + (0.244 − 0.0519i)16-s + (0.758 + 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.164208 - 0.697875i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.164208 - 0.697875i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0523 + 0.998i)T \) |
| 5 | \( 1 + (1.20 - 1.88i)T \) |
| 7 | \( 1 + (-0.823 + 2.51i)T \) |
| good | 3 | \( 1 + (0.158 + 0.128i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (4.28 + 0.911i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.59 + 3.13i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.12 - 1.20i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.644 + 6.13i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (5.32 - 0.279i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.478 - 0.658i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (0.875 + 1.96i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (4.88 - 7.52i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-10.9 - 3.54i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.14 - 5.14i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.179 + 0.466i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-1.91 + 2.36i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (5.51 + 6.12i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (1.23 + 1.11i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (1.25 - 3.27i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-12.1 + 8.79i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.61 + 1.69i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-6.08 + 13.6i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-7.78 + 1.23i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (2.00 - 2.22i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-9.90 - 1.56i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.97447352497791110208646211509, −10.43788675054006754624639800391, −9.567636154999673652608537545374, −8.055605125371535414888293016604, −7.56274708333197450875140207837, −6.26658151321016606812493660825, −4.91344417845436477742848042178, −3.62176201304653677979425423526, −2.75501033995479605733207944293, −0.49610368408848332259000542041,
2.17540173613874137228653319327, 4.14139238531478748435080463920, 5.26544021929230551221162455033, 5.68150045132225509629149031978, 7.55098469161692268510041689085, 7.942971339930223665732091864090, 8.878099572142757960605538556824, 9.866094930475126449969670415051, 10.94562101051338524722795942523, 12.27392421788919938209993889966
