| L(s) = 1 | + (−1.62 − 0.586i)3-s + 2.23i·5-s + 2.64·7-s + (2.31 + 1.91i)9-s + 0.359i·11-s + 4.48·13-s + (1.31 − 3.64i)15-s + 7.99i·17-s + (−4.31 − 1.55i)21-s − 5.00·25-s + (−2.64 − 4.47i)27-s − 10.7i·29-s + (0.211 − 0.586i)33-s + 5.91i·35-s + (−7.31 − 2.63i)39-s + ⋯ |
| L(s) = 1 | + (−0.940 − 0.338i)3-s + 0.999i·5-s + 0.999·7-s + (0.770 + 0.637i)9-s + 0.108i·11-s + 1.24·13-s + (0.338 − 0.940i)15-s + 1.93i·17-s + (−0.940 − 0.338i)21-s − 1.00·25-s + (−0.509 − 0.860i)27-s − 1.99i·29-s + (0.0367 − 0.102i)33-s + 0.999i·35-s + (−1.17 − 0.421i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394117358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394117358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.62 + 0.586i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
| good | 11 | \( 1 - 0.359iT - 11T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 - 7.99iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.7iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12.4iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−9.785666255179660013695067362209, −8.375502337729520389366967129928, −7.941319470631330520162094101987, −7.03161451922476469159759385601, −5.99619377427581646602259551416, −5.92391883081966650039530759261, −4.46234561367333301816584611014, −3.80094652073976026407855235833, −2.25283823559362103658619148274, −1.30090949905488026395656000039,
0.69504470409658723085314324581, 1.65256626768934331583023690861, 3.43031792646757550613845348286, 4.48702081439393629386599658987, 5.11693807179973928840146856814, 5.63210393914303154145519589805, 6.76893549291969135575713998311, 7.56357172961317304740932204298, 8.656191627159801339178922824958, 9.047872721127674255582820596296
