L(s) = 1 | + (0.228 + 0.395i)2-s + (0.866 − 0.5i)3-s + (0.895 − 1.55i)4-s + (−2.18 + 0.456i)5-s + (0.395 + 0.228i)6-s + (−0.866 + 1.5i)7-s + 1.73·8-s + (−1 + 1.73i)9-s + (−0.680 − 0.761i)10-s + (−2.29 + 1.32i)11-s − 1.79i·12-s + (3.46 − i)13-s − 0.791·14-s + (−1.66 + 1.49i)15-s + (−1.39 − 2.41i)16-s + (−3.96 − 2.29i)17-s + ⋯ |
L(s) = 1 | + (0.161 + 0.279i)2-s + (0.499 − 0.288i)3-s + (0.447 − 0.775i)4-s + (−0.978 + 0.204i)5-s + (0.161 + 0.0932i)6-s + (−0.327 + 0.566i)7-s + 0.612·8-s + (−0.333 + 0.577i)9-s + (−0.215 − 0.240i)10-s + (−0.690 + 0.398i)11-s − 0.517i·12-s + (0.960 − 0.277i)13-s − 0.211·14-s + (−0.430 + 0.384i)15-s + (−0.348 − 0.604i)16-s + (−0.962 − 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02275 - 0.0253843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02275 - 0.0253843i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.18 - 0.456i)T \) |
| 13 | \( 1 + (-3.46 + i)T \) |
good | 2 | \( 1 + (-0.228 - 0.395i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (0.866 - 1.5i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.29 - 1.32i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.96 + 2.29i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.29 - 3.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.20iT - 31T^{2} \) |
| 37 | \( 1 + (-3.96 - 6.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.29 + 1.32i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (9.16 + 5.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 - 7.58iT - 53T^{2} \) |
| 59 | \( 1 + (12.0 + 6.97i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.708 + 1.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.504 + 0.873i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.08 - 3.51i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 + 6.01T + 83T^{2} \) |
| 89 | \( 1 + (-8.29 + 4.78i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.70 - 9.87i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.21646402826567298278817069145, −13.89773740875203017955071674250, −12.89074929844707762337727772137, −11.37597480694708182434760309246, −10.60413139986680490071109025125, −8.874224455871232030128177799755, −7.70782616002010372482731212264, −6.50253374052466875181641935995, −4.90972446654755545152386280136, −2.67516543151767637733438152719,
3.20822318418770939066641300386, 4.16201228896303372136314573058, 6.63245746008387717216145898156, 8.010867862262315465823982381317, 8.849400630153204233572009066947, 10.70728630722250951887977646561, 11.51723502988010633281204320021, 12.73785212646342377897351992444, 13.59610760463297749621082631833, 15.12786442662525226043117884281