L(s) = 1 | + (0.309 − 0.535i)2-s + (−1.11 + 1.93i)3-s + (0.809 + 1.40i)4-s + 5-s + (0.690 + 1.19i)6-s + (−2.11 − 3.66i)7-s + 2.23·8-s + (−1 − 1.73i)9-s + (0.309 − 0.535i)10-s + (0.118 − 0.204i)11-s − 3.61·12-s + (−1 − 3.46i)13-s − 2.61·14-s + (−1.11 + 1.93i)15-s + (−0.927 + 1.60i)16-s + (1.73 + 3.00i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.378i)2-s + (−0.645 + 1.11i)3-s + (0.404 + 0.700i)4-s + 0.447·5-s + (0.282 + 0.488i)6-s + (−0.800 − 1.38i)7-s + 0.790·8-s + (−0.333 − 0.577i)9-s + (0.0977 − 0.169i)10-s + (0.0355 − 0.0616i)11-s − 1.04·12-s + (−0.277 − 0.960i)13-s − 0.699·14-s + (−0.288 + 0.500i)15-s + (−0.231 + 0.401i)16-s + (0.421 + 0.729i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.881842 + 0.242359i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.881842 + 0.242359i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 + (-0.309 + 0.535i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.11 - 1.93i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.118 + 0.204i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.73 - 3.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.11 + 3.66i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.88 - 3.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.73 + 6.47i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.97 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.11 - 5.40i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-0.354 - 0.613i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.20 + 12.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.35 + 2.34i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.11 - 5.40i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.73 - 3.00i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−15.27203221728130883942385367391, −13.58824312276214592994386588658, −12.85342784185972246918760586522, −11.41531999749598155395538623730, −10.42568381892046713289286935250, −9.884308230922131675060388385853, −7.84312595428948964394421230079, −6.37010337587138669427635071090, −4.59101232351657401245294500722, −3.37901253656634988939893715786,
2.07564039327805673978410564140, 5.40477671216775047552365708067, 6.27573316742424405601050745681, 7.07440738516544523845869778812, 9.015218288918786119772358897591, 10.30938188485024706140476848320, 11.90051847468747981132561748123, 12.41286964325917469070239502382, 13.74268997700811759479443586501, 14.73300176271884016737820839524