| L(s) = 1 | + (−0.453 + 0.891i)2-s + (−1.59 − 1.03i)3-s + (−0.587 − 0.809i)4-s + (0.0650 − 2.23i)5-s + (1.64 − 0.950i)6-s + (0.498 + 1.29i)7-s + (0.987 − 0.156i)8-s + (0.248 + 0.558i)9-s + (1.96 + 1.07i)10-s + (−5.29 + 0.556i)11-s + (0.0994 + 1.89i)12-s + (−0.331 + 6.31i)13-s + (−1.38 − 0.145i)14-s + (−2.41 + 3.49i)15-s + (−0.309 + 0.951i)16-s + (−0.470 − 0.581i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.630i)2-s + (−0.920 − 0.597i)3-s + (−0.293 − 0.404i)4-s + (0.0291 − 0.999i)5-s + (0.671 − 0.387i)6-s + (0.188 + 0.490i)7-s + (0.349 − 0.0553i)8-s + (0.0829 + 0.186i)9-s + (0.620 + 0.339i)10-s + (−1.59 + 0.167i)11-s + (0.0287 + 0.547i)12-s + (−0.0918 + 1.75i)13-s + (−0.369 − 0.0388i)14-s + (−0.624 + 0.902i)15-s + (−0.0772 + 0.237i)16-s + (−0.114 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00426323 - 0.0748816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00426323 - 0.0748816i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 5 | \( 1 + (-0.0650 + 2.23i)T \) |
| 31 | \( 1 + (1.09 - 5.45i)T \) |
| good | 3 | \( 1 + (1.59 + 1.03i)T + (1.22 + 2.74i)T^{2} \) |
| 7 | \( 1 + (-0.498 - 1.29i)T + (-5.20 + 4.68i)T^{2} \) |
| 11 | \( 1 + (5.29 - 0.556i)T + (10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.331 - 6.31i)T + (-12.9 - 1.35i)T^{2} \) |
| 17 | \( 1 + (0.470 + 0.581i)T + (-3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (5.11 + 4.60i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.470 + 2.96i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.0587 + 0.180i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 0.443i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (7.47 + 1.58i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (8.34 - 0.437i)T + (42.7 - 4.49i)T^{2} \) |
| 47 | \( 1 + (-2.63 + 1.34i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-6.48 - 2.49i)T + (39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (-0.0313 - 0.147i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 8.39iT - 61T^{2} \) |
| 67 | \( 1 + (13.1 - 3.51i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-8.50 + 3.78i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.07 + 1.33i)T + (-15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (0.524 - 4.98i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (5.35 + 8.25i)T + (-33.7 + 75.8i)T^{2} \) |
| 89 | \( 1 + (-9.15 + 6.65i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.67 - 0.265i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.38618687585782248237049071829, −10.30483461680560539126375788454, −9.014698762198939084932363596127, −8.491761371994554897486598779123, −7.18677520421709484280698876024, −6.38461750072313053980975516606, −5.27332404189265788516461031062, −4.62546840216337282680050787961, −1.97878147848491546957694798977, −0.06332448087974712035143624863,
2.52528424856903831697518399783, 3.77335378379846620220224557385, 5.17974803795869346092418542316, 6.00767870932492950732123516531, 7.61230080178931446669364396843, 8.178837952998398295926321153009, 10.09208509298421968726496146464, 10.38969722085301421170907840952, 10.84220573127383427642901050804, 11.78080952046183286101856034111
