| L(s) = 1 | + (−0.206 + 1.39i)2-s + (−1.30 − 0.456i)3-s + (−1.91 − 0.578i)4-s + (−3.64 − 0.831i)5-s + (0.907 − 1.72i)6-s + (0.00558 + 0.0115i)7-s + (1.20 − 2.55i)8-s + (−0.854 − 0.681i)9-s + (1.91 − 4.92i)10-s + (−0.482 + 4.28i)11-s + (2.23 + 1.62i)12-s + (−1.91 + 1.52i)13-s + (−0.0173 + 0.00541i)14-s + (4.36 + 2.74i)15-s + (3.32 + 2.21i)16-s + (0.542 − 0.542i)17-s + ⋯ |
| L(s) = 1 | + (−0.146 + 0.989i)2-s + (−0.752 − 0.263i)3-s + (−0.957 − 0.289i)4-s + (−1.62 − 0.371i)5-s + (0.370 − 0.705i)6-s + (0.00211 + 0.00438i)7-s + (0.426 − 0.904i)8-s + (−0.284 − 0.227i)9-s + (0.605 − 1.55i)10-s + (−0.145 + 1.29i)11-s + (0.644 + 0.469i)12-s + (−0.530 + 0.423i)13-s + (−0.00464 + 0.00144i)14-s + (1.12 + 0.708i)15-s + (0.832 + 0.554i)16-s + (0.131 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00376184 - 0.00866840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00376184 - 0.00866840i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.206 - 1.39i)T \) |
| 29 | \( 1 + (5.34 - 0.662i)T \) |
| good | 3 | \( 1 + (1.30 + 0.456i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (3.64 + 0.831i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-0.00558 - 0.0115i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (0.482 - 4.28i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (1.91 - 1.52i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.542 + 0.542i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.93 + 5.53i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (5.63 - 1.28i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-6.05 + 3.80i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (3.39 - 0.382i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-0.383 - 0.383i)T + 41iT^{2} \) |
| 43 | \( 1 + (0.0943 - 0.150i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (8.20 + 0.924i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (-0.266 + 1.16i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 6.21iT - 59T^{2} \) |
| 61 | \( 1 + (-2.56 + 7.32i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (5.52 - 6.92i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-3.36 - 4.21i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (7.79 - 12.4i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (10.6 - 1.19i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (-1.85 + 3.85i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-3.61 - 5.74i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (2.17 + 6.22i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−13.02432813551907303879879123721, −12.13591682968986072780546533730, −11.38801897519146929571528350864, −9.801590158878922362272588098504, −8.556623580926508602564679509460, −7.49128873935476501146363877998, −6.71950813608798507728786245371, −5.11117458958541968765302739579, −4.14222250925184851682358256028, −0.01167694113146081721857231794,
3.18137352028942332907300957467, 4.33239351446538894701885682357, 5.79813850909181496776638127145, 7.82894609170341812805871643163, 8.457758241111381008965255781498, 10.29556992863821108314113129723, 10.90597425026636864309170296731, 11.80206637228862362222190222471, 12.30763534195131040803115261409, 13.81464995136656344837389476164
