L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−1.84 − 1.26i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.410 + 2.19i)10-s + 5.16·11-s + (0.707 − 0.707i)12-s + (0.184 + 0.184i)13-s + (0.410 + 2.19i)15-s − 1.00·16-s + (−0.750 + 0.750i)17-s + (0.707 − 0.707i)18-s − 4.08·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.824 − 0.565i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.129 + 0.695i)10-s + 1.55·11-s + (0.204 − 0.204i)12-s + (0.0512 + 0.0512i)13-s + (0.106 + 0.567i)15-s − 0.250·16-s + (−0.182 + 0.182i)17-s + (0.166 − 0.166i)18-s − 0.937·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8843416355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8843416355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.84 + 1.26i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.16T + 11T^{2} \) |
| 13 | \( 1 + (-0.184 - 0.184i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.750 - 0.750i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 + (2.36 - 2.36i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.387iT - 29T^{2} \) |
| 31 | \( 1 - 10.6iT - 31T^{2} \) |
| 37 | \( 1 + (-2.51 - 2.51i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.05iT - 41T^{2} \) |
| 43 | \( 1 + (2.13 - 2.13i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.37 + 7.37i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.15 + 7.15i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.70T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 + (0.0355 + 0.0355i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + (3.22 + 3.22i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.46iT - 79T^{2} \) |
| 83 | \( 1 + (0.409 + 0.409i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + (-1.32 + 1.32i)T - 97iT^{2} \) |
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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
−9.347511518414048057827860664716, −8.603815632352681587454988184791, −8.095242264618718521820843613958, −6.97292418112321095318184373455, −6.49018940997294901544618067221, −5.17110579215407567907108312031, −4.18622165466673857610539942582, −3.47900227680175994371075815816, −1.88859689746743394462673614177, −0.899265460258512422010505912375,
0.62793602423771340582673137973, 2.36507230750464786925223988298, 4.00466718910151633575823508376, 4.20836723089094945584677761259, 5.70622080842397765676351530426, 6.42312547731797826240446998314, 7.07270820015044396325750952612, 7.942033392827959747072192531560, 8.825219888415518219489412974740, 9.428537370659390419033818841494