L(s) = 1 | + (−1.35 − 0.780i)2-s + (−0.5 + 0.866i)3-s + (0.219 + 0.379i)4-s + 3.56i·5-s + (1.35 − 0.780i)6-s + (0.486 − 0.280i)7-s + 2.43i·8-s + (−0.499 − 0.866i)9-s + (2.78 − 4.81i)10-s + (1.73 + i)11-s − 0.438·12-s − 0.876·14-s + (−3.08 − 1.78i)15-s + (2.34 − 4.05i)16-s + (−0.780 − 1.35i)17-s + 1.56i·18-s + ⋯ |
L(s) = 1 | + (−0.956 − 0.552i)2-s + (−0.288 + 0.499i)3-s + (0.109 + 0.189i)4-s + 1.59i·5-s + (0.552 − 0.318i)6-s + (0.183 − 0.106i)7-s + 0.862i·8-s + (−0.166 − 0.288i)9-s + (0.879 − 1.52i)10-s + (0.522 + 0.301i)11-s − 0.126·12-s − 0.234·14-s + (−0.796 − 0.459i)15-s + (0.585 − 1.01i)16-s + (−0.189 − 0.327i)17-s + 0.368i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.166259 + 0.397773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.166259 + 0.397773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.35 + 0.780i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.56iT - 5T^{2} \) |
| 7 | \( 1 + (-0.486 + 0.280i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.780 + 1.35i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.16 - 3.56i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.56iT - 31T^{2} \) |
| 37 | \( 1 + (6.54 + 3.78i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.35 + 0.780i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.28 - 3.95i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 + 0.684T + 53T^{2} \) |
| 59 | \( 1 + (2.49 - 1.43i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.95 - 2.28i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (12.1 - 7i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 - 0.876iT - 83T^{2} \) |
| 89 | \( 1 + (4.22 + 2.43i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.41 + 4.28i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.78960381804233993075499579911, −10.59911580052115887803855827151, −9.665373803782442493535356216590, −8.875342156189041009975570503285, −7.74989104111435542896793802455, −6.75598958321425859481217298175, −5.82887978600494103529964122551, −4.40726333376805503480558811124, −3.13558154817047365711907908753, −1.90694529078833871603736934441,
0.36701880213898301268550548762, 1.68909613561630295014376903440, 3.95328059014769722412436031278, 5.01329917265759732932361113633, 6.15978317691766940834864605552, 7.06540268054067223988937769766, 8.187491536796611364333417553443, 8.661011770969954704861224602356, 9.255447071811144264103564732043, 10.36400819505776333580002306686