L(s) = 1 | + (0.928 + 0.928i)2-s + (−1.37 − 1.05i)3-s − 0.276i·4-s + (2.12 + 2.12i)5-s + (−0.293 − 2.25i)6-s + (−2.06 − 2.06i)7-s + (2.11 − 2.11i)8-s + (0.767 + 2.90i)9-s + 3.94i·10-s + (1.88 − 1.88i)11-s + (−0.291 + 0.379i)12-s − 3.83i·14-s + (−0.671 − 5.16i)15-s + 3.37·16-s − 0.198·17-s + (−1.98 + 3.40i)18-s + ⋯ |
L(s) = 1 | + (0.656 + 0.656i)2-s + (−0.792 − 0.610i)3-s − 0.138i·4-s + (0.950 + 0.950i)5-s + (−0.119 − 0.920i)6-s + (−0.780 − 0.780i)7-s + (0.747 − 0.747i)8-s + (0.255 + 0.966i)9-s + 1.24i·10-s + (0.568 − 0.568i)11-s + (−0.0842 + 0.109i)12-s − 1.02i·14-s + (−0.173 − 1.33i)15-s + 0.842·16-s − 0.0482·17-s + (−0.466 + 0.802i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79193 - 0.236084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79193 - 0.236084i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.37 + 1.05i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.928 - 0.928i)T + 2iT^{2} \) |
| 5 | \( 1 + (-2.12 - 2.12i)T + 5iT^{2} \) |
| 7 | \( 1 + (2.06 + 2.06i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.88 + 1.88i)T - 11iT^{2} \) |
| 17 | \( 1 + 0.198T + 17T^{2} \) |
| 19 | \( 1 + (-3.75 + 3.75i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 + 3.73iT - 29T^{2} \) |
| 31 | \( 1 + (0.550 - 0.550i)T - 31iT^{2} \) |
| 37 | \( 1 + (-3.60 - 3.60i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.69 + 2.69i)T + 41iT^{2} \) |
| 43 | \( 1 - 11.6iT - 43T^{2} \) |
| 47 | \( 1 + (-4.29 + 4.29i)T - 47iT^{2} \) |
| 53 | \( 1 - 13.6iT - 53T^{2} \) |
| 59 | \( 1 + (2.36 - 2.36i)T - 59iT^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 + (5.33 - 5.33i)T - 67iT^{2} \) |
| 71 | \( 1 + (3.78 + 3.78i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.46 + 3.46i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + (1.84 + 1.84i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.776 - 0.776i)T - 89iT^{2} \) |
| 97 | \( 1 + (8.60 - 8.60i)T - 97iT^{2} \) |
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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
−10.79090756131629348014210424809, −10.19929229634608327153552639053, −9.344516899359635426674639251194, −7.52144275647243516642208017625, −6.84536238698394073635330151744, −6.29103767647957363765158690278, −5.63581534618560025286642364462, −4.43429388668340967861211594193, −2.93592733949866323743869805305, −1.12555489739670202746274407457,
1.62849816984869880130125900883, 3.15179902007895212774299644546, 4.27320123517215027081174202752, 5.27492414129221993127854834204, 5.80284436050016228164960495061, 7.05358218465488326408662529204, 8.666089981826169615316495249256, 9.421419992869075759833900968232, 10.03681053908892688070753419733, 11.13548340639462754616006905889