| L(s) = 1 | + (−1.10 − 0.887i)2-s + (0.426 + 1.95i)4-s + i·5-s + (0.391 + 2.61i)7-s + (1.26 − 2.53i)8-s + (0.887 − 1.10i)10-s + 0.770i·11-s + 5.60i·13-s + (1.88 − 3.22i)14-s + (−3.63 + 1.66i)16-s − 0.503i·17-s + 1.63·19-s + (−1.95 + 0.426i)20-s + (0.683 − 0.848i)22-s − 1.42i·23-s + ⋯ |
| L(s) = 1 | + (−0.778 − 0.627i)2-s + (0.213 + 0.977i)4-s + 0.447i·5-s + (0.148 + 0.988i)7-s + (0.446 − 0.894i)8-s + (0.280 − 0.348i)10-s + 0.232i·11-s + 1.55i·13-s + (0.504 − 0.863i)14-s + (−0.909 + 0.416i)16-s − 0.122i·17-s + 0.375·19-s + (−0.436 + 0.0953i)20-s + (0.145 − 0.180i)22-s − 0.297i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7473104447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7473104447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.10 + 0.887i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.391 - 2.61i)T \) |
| good | 11 | \( 1 - 0.770iT - 11T^{2} \) |
| 13 | \( 1 - 5.60iT - 13T^{2} \) |
| 17 | \( 1 + 0.503iT - 17T^{2} \) |
| 19 | \( 1 - 1.63T + 19T^{2} \) |
| 23 | \( 1 + 1.42iT - 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 5.07iT - 41T^{2} \) |
| 43 | \( 1 + 9.06iT - 43T^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 + 0.455T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 3.32iT - 61T^{2} \) |
| 67 | \( 1 + 8.70iT - 67T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 + 2.29iT - 73T^{2} \) |
| 79 | \( 1 - 2.56iT - 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 2.98iT - 89T^{2} \) |
| 97 | \( 1 - 15.5iT - 97T^{2} \) |
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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
−9.628631970642491797748026773684, −9.351935656196986493277645340307, −8.530800445258446211441292954921, −7.58339844141764894483694354662, −6.86207745252915784991340665171, −5.92078230739944663804276643560, −4.63789811442500175250515605889, −3.61916320645312621569521749742, −2.48937294032793724057958099563, −1.70605345317545136382271081494,
0.41172384659962475236013019822, 1.54644250730831161165899076144, 3.19773250716726127003610201420, 4.45966093241590432151323141853, 5.44730975972034635518093148382, 6.08733229489478423127072242975, 7.35572785889526016922377596901, 7.70650423348464568330880808399, 8.497325930289528444839366501823, 9.467518292688418538725397053764
