L(s) = 1 | + (2 + 1.73i)7-s + 1.73i·13-s + (3 + 1.73i)19-s + (2.5 + 4.33i)25-s + (−1.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + 13·43-s + (1.00 + 6.92i)49-s + (13.5 + 7.79i)61-s + (−5.5 − 9.52i)67-s + (−12 + 6.92i)73-s + (6.5 − 11.2i)79-s + (−2.99 + 3.46i)91-s + 5.19i·97-s + (−16.5 − 9.52i)103-s + ⋯ |
L(s) = 1 | + (0.755 + 0.654i)7-s + 0.480i·13-s + (0.688 + 0.397i)19-s + (0.5 + 0.866i)25-s + (−0.269 + 0.155i)31-s + (−0.0821 + 0.142i)37-s + 1.98·43-s + (0.142 + 0.989i)49-s + (1.72 + 0.997i)61-s + (−0.671 − 1.16i)67-s + (−1.40 + 0.810i)73-s + (0.731 − 1.26i)79-s + (−0.314 + 0.363i)91-s + 0.527i·97-s + (−1.62 − 0.938i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53238 + 0.585840i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53238 + 0.585840i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 13T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-13.5 - 7.79i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.19iT - 97T^{2} \) |
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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
−10.53221965134939149904171381490, −9.427848159944917642142647213695, −8.825278386768614158005947042004, −7.86676582017030471190609342934, −7.06981337675508768949455224979, −5.87776417435448985938693139540, −5.13269741258244896107698922896, −4.05741181497139106535100787429, −2.74675457327283956911640880157, −1.47871982250095917530485152702,
0.959911025670626443655042082832, 2.50729415977030945124102090545, 3.83043812705157068655464957497, 4.79236386620650775355487539075, 5.71067950860799550435907496887, 6.89937837502719350405088009205, 7.66345269815501757713572860055, 8.444822191514103534905182382007, 9.419036510147655251564257092153, 10.37059750720125497271804065196