| L(s) = 1 | + (−1.26 + 1.84i)5-s + 3.12i·7-s + 11-s + 0.710i·13-s − 7.65i·17-s − 1.02·19-s − 6.36i·23-s + (−1.81 − 4.65i)25-s − 0.234·29-s − 6.88·31-s + (−5.77 − 3.94i)35-s − 8.84i·37-s + 11.0·41-s − 6.28i·43-s − 3.15i·47-s + ⋯ |
| L(s) = 1 | + (−0.564 + 0.825i)5-s + 1.18i·7-s + 0.301·11-s + 0.196i·13-s − 1.85i·17-s − 0.235·19-s − 1.32i·23-s + (−0.363 − 0.931i)25-s − 0.0434·29-s − 1.23·31-s + (−0.976 − 0.667i)35-s − 1.45i·37-s + 1.72·41-s − 0.957i·43-s − 0.459i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.254099522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254099522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.26 - 1.84i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3.12iT - 7T^{2} \) |
| 13 | \( 1 - 0.710iT - 13T^{2} \) |
| 17 | \( 1 + 7.65iT - 17T^{2} \) |
| 19 | \( 1 + 1.02T + 19T^{2} \) |
| 23 | \( 1 + 6.36iT - 23T^{2} \) |
| 29 | \( 1 + 0.234T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 8.84iT - 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 6.28iT - 43T^{2} \) |
| 47 | \( 1 + 3.15iT - 47T^{2} \) |
| 53 | \( 1 - 8.79iT - 53T^{2} \) |
| 59 | \( 1 - 8.01T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 4.21iT - 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 5.47iT - 73T^{2} \) |
| 79 | \( 1 - 1.50T + 79T^{2} \) |
| 83 | \( 1 + 8.70iT - 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 + 17.8iT - 97T^{2} \) |
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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
−8.500862104030783966189806008645, −7.43741939954125223515232345225, −7.09826667229669765028570438034, −6.16095256688566250823955591289, −5.50788666211757093467209597230, −4.56654254460117817164858187147, −3.72963928759149152589977376434, −2.67757905296130804655499703046, −2.26541971433270581174153569977, −0.43183182134641178541071802495,
0.982647284149790886781367879995, 1.76091786270180830376772280644, 3.42625216275978042920187260918, 3.91833009551222517443305690062, 4.59006632553667391901618487123, 5.53319187594069280193363213036, 6.32495464924314219399901250655, 7.20961304963735295382578053611, 7.914103497649092749273109546709, 8.303796876443072630377583720460
