| L(s) = 1 | + (−0.112 + 0.0331i)2-s + (−0.654 − 0.755i)3-s + (−1.67 + 1.07i)4-s + (0.142 + 0.989i)5-s + (0.0988 + 0.0635i)6-s + (1.17 − 2.56i)7-s + (0.306 − 0.354i)8-s + (−0.142 + 0.989i)9-s + (−0.0488 − 0.106i)10-s + (0.434 + 0.127i)11-s + (1.90 + 0.559i)12-s + (2.58 + 5.65i)13-s + (−0.0472 + 0.328i)14-s + (0.654 − 0.755i)15-s + (1.62 − 3.56i)16-s + (5.98 + 3.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.0797 + 0.0234i)2-s + (−0.378 − 0.436i)3-s + (−0.835 + 0.536i)4-s + (0.0636 + 0.442i)5-s + (0.0403 + 0.0259i)6-s + (0.443 − 0.970i)7-s + (0.108 − 0.125i)8-s + (−0.0474 + 0.329i)9-s + (−0.0154 − 0.0338i)10-s + (0.130 + 0.0384i)11-s + (0.550 + 0.161i)12-s + (0.716 + 1.56i)13-s + (−0.0126 + 0.0877i)14-s + (0.169 − 0.195i)15-s + (0.406 − 0.890i)16-s + (1.45 + 0.932i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.985105 + 0.268871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.985105 + 0.268871i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (0.0730 - 4.79i)T \) |
| good | 2 | \( 1 + (0.112 - 0.0331i)T + (1.68 - 1.08i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.56i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.434 - 0.127i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.58 - 5.65i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.98 - 3.84i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.224 + 0.144i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.74 - 2.40i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.12 + 3.60i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.542 + 3.76i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.203 + 1.41i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.72 + 8.91i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + (4.14 - 9.08i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.75 - 3.84i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.77 + 2.04i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (5.72 - 1.67i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (4.08 - 1.19i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.10 + 3.92i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.00 + 10.9i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (2.07 - 14.3i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (7.11 + 8.21i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.14 - 7.97i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.67782658402690196939431729407, −10.69213581486176679381230126170, −9.793622603061353611484051784256, −8.683514596418448824769913695627, −7.72411381306883964469620956358, −6.99025993214446535261739780395, −5.77803349514414949199279679191, −4.40368292448328545798047539749, −3.56201963073466245546351152243, −1.39547391272471680043964943597,
0.970660095004951683114454126853, 3.16294071606359637190232588649, 4.75784721797072899308715601991, 5.35458917552609568039953863544, 6.17792294063350976189683197639, 8.126704776666388708851079606838, 8.579979943194838697460953343163, 9.756731762451740372748875434414, 10.25816790309370366597734075310, 11.42877944610719922993800894762
