L(s) = 1 | + (0.279 − 0.384i)2-s + (0.261 + 0.190i)3-s + (0.548 + 1.68i)4-s + (0.00765 + 0.0235i)5-s + (0.146 − 0.0475i)6-s + (−2.52 − 3.47i)7-s + (1.70 + 0.554i)8-s + (−0.894 − 2.75i)9-s + (0.0111 + 0.00363i)10-s + 5.23i·11-s + (−0.177 + 0.545i)12-s − 4.67·13-s − 2.04·14-s + (−0.00247 + 0.00762i)15-s + (−2.18 + 1.58i)16-s + (6.11 − 1.98i)17-s + ⋯ |
L(s) = 1 | + (0.197 − 0.271i)2-s + (0.151 + 0.109i)3-s + (0.274 + 0.843i)4-s + (0.00342 + 0.0105i)5-s + (0.0597 − 0.0194i)6-s + (−0.954 − 1.31i)7-s + (0.603 + 0.195i)8-s + (−0.298 − 0.917i)9-s + (0.00354 + 0.00115i)10-s + 1.57i·11-s + (−0.0512 + 0.157i)12-s − 1.29·13-s − 0.545·14-s + (−0.000639 + 0.00196i)15-s + (−0.545 + 0.396i)16-s + (1.48 − 0.481i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.975762 - 0.00912798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.975762 - 0.00912798i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 + (3.53 + 6.96i)T \) |
good | 2 | \( 1 + (-0.279 + 0.384i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.261 - 0.190i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.00765 - 0.0235i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (2.52 + 3.47i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 - 5.23iT - 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + (-6.11 + 1.98i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.444 + 0.323i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.83 - 0.922i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 - 3.09iT - 29T^{2} \) |
| 31 | \( 1 + (-1.35 - 1.86i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.76 - 3.80i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-5.91 + 4.29i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (3.17 + 1.03i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + (-1.61 - 0.524i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (4.91 - 6.77i)T + (-18.2 - 56.1i)T^{2} \) |
| 67 | \( 1 + (5.02 - 1.63i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (8.01 + 2.60i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.62 + 8.08i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.24 - 2.67i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.02 - 2.20i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.24 - 3.09i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.159 + 0.115i)T + (29.9 - 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.94771408458435037622760310218, −13.90024132816602151539920502944, −12.46834664661349595190525024730, −12.19382385623645805510435341210, −10.33424785459420140449305803699, −9.538479913582784930404315280271, −7.59218412362502445940752279320, −6.86425873055317073554284222225, −4.41627503291210573899992858802, −3.08246231793195354184388249713,
2.72406887281959213704636539487, 5.38643892840875761509012138385, 6.12716203106126679243216412650, 7.87855792819405271785193330688, 9.301714314031079072598511573722, 10.43149802482416774241279698981, 11.73735789668509476222410225353, 12.97045765361576498589678681121, 14.17600815485692434888160944748, 14.95719551257055633305698193406