L(s) = 1 | + (0.5 − 0.866i)2-s + (0.686 − 2.56i)3-s + (−0.499 − 0.866i)4-s + (−1.84 + 1.25i)5-s + (−1.87 − 1.87i)6-s + (0.524 − 1.95i)7-s − 0.999·8-s + (−3.50 − 2.02i)9-s + (0.166 + 2.22i)10-s + 1.12i·11-s + (−2.56 + 0.686i)12-s + (−2.37 − 4.12i)13-s + (−1.43 − 1.43i)14-s + (1.95 + 5.60i)15-s + (−0.5 + 0.866i)16-s + (0.528 + 0.304i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.396 − 1.47i)3-s + (−0.249 − 0.433i)4-s + (−0.826 + 0.562i)5-s + (−0.766 − 0.766i)6-s + (0.198 − 0.739i)7-s − 0.353·8-s + (−1.16 − 0.673i)9-s + (0.0525 + 0.705i)10-s + 0.338i·11-s + (−0.739 + 0.198i)12-s + (−0.659 − 1.14i)13-s + (−0.382 − 0.382i)14-s + (0.505 + 1.44i)15-s + (−0.125 + 0.216i)16-s + (0.128 + 0.0739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.137403 - 1.39489i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.137403 - 1.39489i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (1.84 - 1.25i)T \) |
| 37 | \( 1 + (-4.85 + 3.67i)T \) |
good | 3 | \( 1 + (-0.686 + 2.56i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-0.524 + 1.95i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 - 1.12iT - 11T^{2} \) |
| 13 | \( 1 + (2.37 + 4.12i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.528 - 0.304i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.968 + 0.259i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 5.88T + 23T^{2} \) |
| 29 | \( 1 + (-2.37 - 2.37i)T + 29iT^{2} \) |
| 31 | \( 1 + (3.83 - 3.83i)T - 31iT^{2} \) |
| 41 | \( 1 + (-7.57 + 4.37i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 + (-4.16 - 4.16i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.917 + 3.42i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.81 + 6.77i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (13.4 + 3.60i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-14.0 - 3.75i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.54 + 6.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.20 + 8.20i)T + 73iT^{2} \) |
| 79 | \( 1 + (-4.51 - 1.20i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (0.924 + 3.44i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-13.2 + 3.55i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−11.01491110815263696803527392618, −10.46820620425621469306572283936, −9.011098295134260608430394058677, −7.74359479983343381422076537335, −7.42551939539688601407153577218, −6.39823083924605014672535678483, −4.87377061747192889729961867056, −3.46357518513701517600991697144, −2.46190135682844976209466728264, −0.845278320603483256831348238766,
2.89673219238015931195200301451, 4.16979849109866009857392240442, 4.69815102750783855626899536155, 5.71693792823982842736413682515, 7.21332587223060126024373832531, 8.349938394727327644285043874557, 9.020276561213244277707391773625, 9.636618304790533874123535940968, 11.03327034124782536935211163382, 11.73833110397170268186875617161