| L(s) = 1 | + (−0.0523 − 0.998i)2-s + (−1.68 − 1.36i)3-s + (−0.994 + 0.104i)4-s + (2.17 + 0.499i)5-s + (−1.27 + 1.75i)6-s + (−2.26 − 1.36i)7-s + (0.156 + 0.987i)8-s + (0.356 + 1.67i)9-s + (0.384 − 2.20i)10-s + (−4.13 − 0.878i)11-s + (1.82 + 1.18i)12-s + (−1.47 − 2.88i)13-s + (−1.24 + 2.33i)14-s + (−2.99 − 3.82i)15-s + (0.978 − 0.207i)16-s + (−2.51 − 0.963i)17-s + ⋯ |
| L(s) = 1 | + (−0.0370 − 0.706i)2-s + (−0.974 − 0.788i)3-s + (−0.497 + 0.0522i)4-s + (0.974 + 0.223i)5-s + (−0.520 + 0.717i)6-s + (−0.856 − 0.516i)7-s + (0.0553 + 0.349i)8-s + (0.118 + 0.558i)9-s + (0.121 − 0.696i)10-s + (−1.24 − 0.264i)11-s + (0.525 + 0.341i)12-s + (−0.408 − 0.801i)13-s + (−0.332 + 0.623i)14-s + (−0.773 − 0.986i)15-s + (0.244 − 0.0519i)16-s + (−0.608 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.131467 + 0.392653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131467 + 0.392653i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.17 - 0.499i)T \) |
| 7 | \( 1 + (2.26 + 1.36i)T \) |
| good | 3 | \( 1 + (1.68 + 1.36i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (4.13 + 0.878i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.47 + 2.88i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (2.51 + 0.963i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.765 - 7.28i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.420 - 0.0220i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-1.60 - 2.20i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.29 + 5.15i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.14 + 6.38i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (8.21 + 2.66i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (5.92 + 5.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.06 + 2.78i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-3.55 + 4.38i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-1.52 - 1.68i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (6.08 + 5.47i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.21 + 8.36i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-12.4 + 9.05i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (12.0 - 7.84i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (0.551 - 1.23i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-15.4 + 2.43i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-5.00 + 5.55i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (13.0 + 2.06i)T + (92.2 + 29.9i)T^{2} \) |
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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
−10.70738251093723798311173585498, −10.37014522435817171770056613393, −9.451640896712106166341755955675, −8.011214265132695055668088706150, −6.93388055209041532122437442927, −5.93369647876849130780430360775, −5.25641539644828084034460853329, −3.40794731076427289951664718838, −2.05661671127554959427466120125, −0.30391913153978530252926808962,
2.60498423843884149202552915195, 4.66432272552524277775701072504, 5.15166997364201499474761852911, 6.20128290229760344168458840277, 6.87889586994228382913702239953, 8.495708187435949161417744419866, 9.456368476712853812713202327921, 10.03123044774981950447325023482, 10.90014710464042402349650177908, 12.02644500504704636434463460504
