L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.66 + 1.49i)5-s + (−0.170 + 0.170i)7-s + (0.707 − 0.707i)8-s + (−0.120 − 2.23i)10-s − 3.43·11-s + (4.54 − 4.54i)13-s + 0.240·14-s − 1.00·16-s + (−0.537 + 0.537i)17-s + (2.42 − 3.62i)19-s + (−1.49 + 1.66i)20-s + (2.42 + 2.42i)22-s + (−5.15 − 5.15i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.744 + 0.668i)5-s + (−0.0643 + 0.0643i)7-s + (0.250 − 0.250i)8-s + (−0.0380 − 0.706i)10-s − 1.03·11-s + (1.25 − 1.25i)13-s + 0.0643·14-s − 0.250·16-s + (−0.130 + 0.130i)17-s + (0.556 − 0.830i)19-s + (−0.334 + 0.372i)20-s + (0.517 + 0.517i)22-s + (−1.07 − 1.07i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.393130563\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.393130563\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1.66 - 1.49i)T \) |
| 19 | \( 1 + (-2.42 + 3.62i)T \) |
good | 7 | \( 1 + (0.170 - 0.170i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 + (-4.54 + 4.54i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.537 - 0.537i)T - 17iT^{2} \) |
| 23 | \( 1 + (5.15 + 5.15i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 5.37iT - 31T^{2} \) |
| 37 | \( 1 + (5.54 + 5.54i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.68iT - 41T^{2} \) |
| 43 | \( 1 + (-2.29 - 2.29i)T + 43iT^{2} \) |
| 47 | \( 1 + (-9.57 + 9.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.93 + 1.93i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 + (-0.481 - 0.481i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.93iT - 71T^{2} \) |
| 73 | \( 1 + (-8.74 - 8.74i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.15T + 79T^{2} \) |
| 83 | \( 1 + (-9.97 - 9.97i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \) |
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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
−9.304296286755974188787977988291, −8.409609771839241599568604684126, −7.83347758051250537538592422993, −6.83173175691019052612821934195, −5.97444269643236993331181794150, −5.25222632085809129110556121810, −3.89684371407431079851532533323, −2.88288555717907046750875649287, −2.25070361136911718592159766402, −0.70277546661185463656287733277,
1.18891436115489559784152402604, 2.12385495495238815501224928426, 3.63362273902904771627688465524, 4.74411163848969219463337811106, 5.59271707509867225092668025548, 6.19686381382453201604714724607, 7.09466611185242472011703479159, 8.063262475056887108018169265874, 8.670778639785826314035385820800, 9.336229537858985788865403927503