L(s) = 1 | + (4.72 − 2.15i)3-s + (1.13 + 7.91i)4-s + (11.6 + 10.0i)7-s + (17.6 − 20.4i)9-s + (22.4 + 34.9i)12-s + (6.52 + 10.1i)13-s + (−61.4 + 18.0i)16-s + (108. + 124. i)19-s + (76.5 + 22.4i)21-s + (−105. + 67.5i)25-s + (39.5 − 134. i)27-s + (−66.4 + 103. i)28-s + (127. − 198. i)31-s + (181. + 116. i)36-s + 143.·37-s + ⋯ |
L(s) = 1 | + (0.909 − 0.415i)3-s + (0.142 + 0.989i)4-s + (0.626 + 0.543i)7-s + (0.654 − 0.755i)9-s + (0.540 + 0.841i)12-s + (0.139 + 0.216i)13-s + (−0.959 + 0.281i)16-s + (1.30 + 1.50i)19-s + (0.795 + 0.233i)21-s + (−0.841 + 0.540i)25-s + (0.281 − 0.959i)27-s + (−0.448 + 0.697i)28-s + (0.740 − 1.15i)31-s + (0.841 + 0.540i)36-s + 0.637·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.46428 + 0.887514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46428 + 0.887514i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.72 + 2.15i)T \) |
| 67 | \( 1 + (-193. + 513. i)T \) |
good | 2 | \( 1 + (-1.13 - 7.91i)T^{2} \) |
| 5 | \( 1 + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (-11.6 - 10.0i)T + (48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (-6.52 - 10.1i)T + (-912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-108. - 124. i)T + (-976. + 6.78e3i)T^{2} \) |
| 23 | \( 1 + (7.96e3 - 9.19e3i)T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + (-127. + 198. i)T + (-1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-96.5 - 13.8i)T + (7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + (6.79e4 - 7.84e4i)T^{2} \) |
| 53 | \( 1 + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (156. - 531. i)T + (-1.90e5 - 1.22e5i)T^{2} \) |
| 71 | \( 1 + (3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (1.19e3 + 351. i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (754. + 1.17e3i)T + (-2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + 1.77e3iT - 9.12e5T^{2} \) |
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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
−12.04535490328697813685676911863, −11.57599029450035145695544509356, −9.886022045974297538408445530002, −8.891130740696405260538033963917, −7.963986625241423713847589223312, −7.44027094431144237997940026543, −5.95399873626036341225709415916, −4.19357240487032058596636134192, −3.06810282134756963647375859535, −1.77087436685935864981232633078,
1.17950517494264521580499045623, 2.72159204847951802476787886950, 4.35598088846051187259204263038, 5.30077545943877556901095558762, 6.86819770905064173115550806280, 7.893362268911845402201503254980, 9.062050812822950387041777913071, 9.906947794607944767195770401437, 10.74220560566686446754589329728, 11.61319739384845226843321472284