| L(s) = 1 | + (−0.462 − 1.66i)3-s − 5-s + (−1.62 − 2.08i)7-s + (−2.57 + 1.54i)9-s − 0.196i·11-s + 2.37i·13-s + (0.462 + 1.66i)15-s − 3.36·17-s − 2.89i·19-s + (−2.72 + 3.68i)21-s + 5.80i·23-s + 25-s + (3.76 + 3.57i)27-s + 5.73i·29-s − 4.66i·31-s + ⋯ |
| L(s) = 1 | + (−0.267 − 0.963i)3-s − 0.447·5-s + (−0.615 − 0.788i)7-s + (−0.857 + 0.514i)9-s − 0.0591i·11-s + 0.659i·13-s + (0.119 + 0.430i)15-s − 0.817·17-s − 0.663i·19-s + (−0.594 + 0.803i)21-s + 1.20i·23-s + 0.200·25-s + (0.725 + 0.688i)27-s + 1.06i·29-s − 0.838i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5671150958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5671150958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.462 + 1.66i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (1.62 + 2.08i)T \) |
| good | 11 | \( 1 + 0.196iT - 11T^{2} \) |
| 13 | \( 1 - 2.37iT - 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 + 2.89iT - 19T^{2} \) |
| 23 | \( 1 - 5.80iT - 23T^{2} \) |
| 29 | \( 1 - 5.73iT - 29T^{2} \) |
| 31 | \( 1 + 4.66iT - 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 - 0.444T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 - 2.24iT - 53T^{2} \) |
| 59 | \( 1 - 4.10T + 59T^{2} \) |
| 61 | \( 1 - 1.24iT - 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 8.21iT - 71T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 - 9.71T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 - 2.30T + 89T^{2} \) |
| 97 | \( 1 + 6.59iT - 97T^{2} \) |
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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
−9.284655154904928804874677845716, −8.690285673091027449982757404835, −7.56334296895961180443808002965, −7.15005718126269115296410509154, −6.48104134186597906133779062181, −5.53996197265902689830876119021, −4.44955293823038122306321701053, −3.50822322123027103700800932838, −2.36979557691855465531006481916, −1.06835902743799567375973631764,
0.25645620744159193699468686600, 2.41555493715124815076194445453, 3.32991950838985110101237212910, 4.21629355587796457787966768872, 5.08437483554706816905713064853, 5.95590713130678187339819208268, 6.59321064363213535250213920322, 7.81232732533167357498816284665, 8.692911169656877201505858343096, 9.140611144718964879590836674732
