L(s) = 1 | + (0.478 + 1.66i)3-s + 2.74·5-s + (1.70 − 2.02i)7-s + (−2.54 + 1.59i)9-s + 0.418i·11-s + (−1.32 − 0.765i)13-s + (1.31 + 4.56i)15-s + (1.95 − 3.38i)17-s + (−5.11 + 2.95i)19-s + (4.18 + 1.86i)21-s + 8.92i·23-s + 2.52·25-s + (−3.86 − 3.47i)27-s + (6.00 − 3.46i)29-s + (−3.05 + 1.76i)31-s + ⋯ |
L(s) = 1 | + (0.276 + 0.961i)3-s + 1.22·5-s + (0.644 − 0.764i)7-s + (−0.847 + 0.530i)9-s + 0.126i·11-s + (−0.367 − 0.212i)13-s + (0.338 + 1.17i)15-s + (0.473 − 0.820i)17-s + (−1.17 + 0.678i)19-s + (0.913 + 0.407i)21-s + 1.86i·23-s + 0.505·25-s + (−0.744 − 0.667i)27-s + (1.11 − 0.643i)29-s + (−0.548 + 0.316i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52700 + 0.544204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52700 + 0.544204i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.478 - 1.66i)T \) |
| 7 | \( 1 + (-1.70 + 2.02i)T \) |
good | 5 | \( 1 - 2.74T + 5T^{2} \) |
| 11 | \( 1 - 0.418iT - 11T^{2} \) |
| 13 | \( 1 + (1.32 + 0.765i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.11 - 2.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 8.92iT - 23T^{2} \) |
| 29 | \( 1 + (-6.00 + 3.46i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.05 - 1.76i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.54 + 7.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.06 - 1.84i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.77 + 10.0i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.885 - 1.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.39 - 1.96i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.02 + 3.51i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.61 + 0.932i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.38 - 11.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (-1.65 - 0.952i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.433 + 0.751i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.45 + 5.99i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.88 - 8.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.200 - 0.115i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.02527146067836366265443553804, −10.88193256649387284372289978807, −10.13412655287246162341409979160, −9.535543358983032556967021585031, −8.417093427326245059452371275273, −7.27916621861752830548173668082, −5.76206342114621064001003475360, −4.94666737785745907883076237133, −3.65129886167901715786712180661, −2.03184608511635421057746453389,
1.73604667345096736976918084728, 2.69236532658584826482278878805, 4.87397374842858634448142351721, 6.07302829371557532368346564733, 6.71989863370938716635595396598, 8.293378691572473002118656232219, 8.745654315705607774075133652186, 9.985435816762939639147587868715, 11.04087537707768793587292147916, 12.23482736743453083954741359916