| L(s) = 1 | + 3-s − 2i·5-s − 2i·7-s + 9-s + 4i·11-s + (3 + 2i)13-s − 2i·15-s + 6·17-s − 2i·19-s − 2i·21-s + 4·23-s + 25-s + 27-s − 6·29-s + 2i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894i·5-s − 0.755i·7-s + 0.333·9-s + 1.20i·11-s + (0.832 + 0.554i)13-s − 0.516i·15-s + 1.45·17-s − 0.458i·19-s − 0.436i·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s − 1.11·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.513204143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.513204143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.800231580658659328363023192315, −8.213538201092825098664994089744, −7.23962035746449246622976859874, −6.89039316028772967501975001159, −5.54737502406741146094560996999, −4.80001896832332784332943775094, −4.03550261496873049419374245494, −3.22856447382543588138950231249, −1.83705105003990006574639166354, −1.00358543246218362034925822145,
1.11790060136723994029831466346, 2.50711032602554537129368615231, 3.26690015409259486113514681698, 3.75025899673249040232327168009, 5.34244430060379398760697898457, 5.84069307961661612167513585185, 6.65856701668066034663351837868, 7.66617087559800084747141272140, 8.177821209662886058827199205650, 8.948100152779595639890198604817
