| L(s) = 1 | + (1.64 − 0.539i)3-s + (1.30 + 1.81i)5-s + (2.19 − 1.47i)7-s + (2.41 − 1.77i)9-s + 0.958i·11-s − 0.157·13-s + (3.12 + 2.29i)15-s + 2.51i·17-s − 1.98i·19-s + (2.82 − 3.60i)21-s − 2.67·23-s + (−1.60 + 4.73i)25-s + (3.02 − 4.22i)27-s + 1.25i·29-s − 8.66i·31-s + ⋯ |
| L(s) = 1 | + (0.950 − 0.311i)3-s + (0.582 + 0.812i)5-s + (0.831 − 0.555i)7-s + (0.806 − 0.591i)9-s + 0.288i·11-s − 0.0436·13-s + (0.806 + 0.591i)15-s + 0.609i·17-s − 0.454i·19-s + (0.617 − 0.786i)21-s − 0.557·23-s + (−0.321 + 0.946i)25-s + (0.582 − 0.813i)27-s + 0.232i·29-s − 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.58948 - 0.0564309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.58948 - 0.0564309i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 + 0.539i)T \) |
| 5 | \( 1 + (-1.30 - 1.81i)T \) |
| 7 | \( 1 + (-2.19 + 1.47i)T \) |
| good | 11 | \( 1 - 0.958iT - 11T^{2} \) |
| 13 | \( 1 + 0.157T + 13T^{2} \) |
| 17 | \( 1 - 2.51iT - 17T^{2} \) |
| 19 | \( 1 + 1.98iT - 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 - 1.25iT - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 2.29iT - 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 - 6.58iT - 43T^{2} \) |
| 47 | \( 1 - 5.60iT - 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.27iT - 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 9.75iT - 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 - 0.517T + 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 - 0.954T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−10.06066731641979541406612506674, −9.464138610363481455575000548158, −8.344778986165073220784725168925, −7.68553813692794580991372905503, −6.90941589711092452790330214022, −6.03008119463204554308287123179, −4.64930627227472480208131111004, −3.66043058122366498110544338731, −2.49836128903172648963540818485, −1.55332495272397963436760727759,
1.52406975322080882284661679210, 2.49504938503724771420864761216, 3.82252080158701566152783472126, 4.92013922938516073863610464953, 5.50770640734503146398843608094, 6.86605421499059859871134237044, 8.064759945698657983711757396471, 8.506012033485108036104429563203, 9.258587617740033622224537935495, 9.994295700681714190274753228380
