| L(s) = 1 | + (−1.59 − 2.75i)5-s + (−0.607 + 1.05i)7-s + (0.312 − 0.540i)11-s + (1.06 + 1.84i)13-s + 1.83·17-s − 7.15·19-s + (0.780 + 1.35i)23-s + (−2.57 + 4.46i)25-s + (4.87 − 8.44i)29-s + (−3.32 − 5.75i)31-s + 3.86·35-s − 6.73·37-s + (5.64 + 9.77i)41-s + (−4.51 + 7.81i)43-s + (−1.36 + 2.35i)47-s + ⋯ |
| L(s) = 1 | + (−0.712 − 1.23i)5-s + (−0.229 + 0.397i)7-s + (0.0941 − 0.163i)11-s + (0.295 + 0.511i)13-s + 0.445·17-s − 1.64·19-s + (0.162 + 0.282i)23-s + (−0.515 + 0.892i)25-s + (0.905 − 1.56i)29-s + (−0.596 − 1.03i)31-s + 0.653·35-s − 1.10·37-s + (0.881 + 1.52i)41-s + (−0.687 + 1.19i)43-s + (−0.198 + 0.343i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5172351867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5172351867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.607 - 1.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.312 + 0.540i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 1.84i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 + 7.15T + 19T^{2} \) |
| 23 | \( 1 + (-0.780 - 1.35i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.87 + 8.44i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.32 + 5.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 + (-5.64 - 9.77i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.51 - 7.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.36 - 2.35i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 + (4.02 + 6.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.79 + 4.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.11T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 + (3.21 - 5.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.27 - 5.67i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 5.02T + 89T^{2} \) |
| 97 | \( 1 + (4.70 - 8.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.601076559550641653056640311634, −8.262224944569414135284218312654, −7.51553384138469959764116440259, −6.34866185746215656371718072252, −5.92883147991801824023178412548, −4.70132523808893705457884229439, −4.38412531462669409219855650338, −3.41633874241712128804047015109, −2.22544352355997440475438081920, −1.07858137546060645679054755060,
0.17260250912173617699149509398, 1.79731281097876250206714929532, 3.02476867201814838153263603587, 3.51961228410654575653896832157, 4.36815897959062755191871327284, 5.39690647989198046070625664437, 6.35926246749946921071526910211, 7.07121594529355350278797909116, 7.33203295680595737244316031545, 8.520750037941383620783242806165
