L(s) = 1 | + (−0.259 + 0.969i)2-s + (−0.965 + 0.258i)3-s + (0.859 + 0.496i)4-s − 1.00i·6-s + (1.06 − 2.42i)7-s + (−2.12 + 2.12i)8-s + (0.866 − 0.499i)9-s + (−1.78 + 3.08i)11-s + (−0.958 − 0.256i)12-s + (2.78 + 2.78i)13-s + (2.07 + 1.65i)14-s + (−0.514 − 0.891i)16-s + (0.135 + 0.506i)17-s + (0.259 + 0.969i)18-s + (2.06 + 3.57i)19-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.685i)2-s + (−0.557 + 0.149i)3-s + (0.429 + 0.248i)4-s − 0.409i·6-s + (0.401 − 0.915i)7-s + (−0.750 + 0.750i)8-s + (0.288 − 0.166i)9-s + (−0.537 + 0.931i)11-s + (−0.276 − 0.0741i)12-s + (0.772 + 0.772i)13-s + (0.554 + 0.443i)14-s + (−0.128 − 0.222i)16-s + (0.0329 + 0.122i)17-s + (0.0612 + 0.228i)18-s + (0.473 + 0.819i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.625784 + 1.01879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.625784 + 1.01879i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.06 + 2.42i)T \) |
good | 2 | \( 1 + (0.259 - 0.969i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.78 - 2.78i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.135 - 0.506i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.06 - 3.57i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.49 - 0.668i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 6.14iT - 29T^{2} \) |
| 31 | \( 1 + (1.71 + 0.988i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0409 + 0.152i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 8.28iT - 41T^{2} \) |
| 43 | \( 1 + (9.01 - 9.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.19 - 1.39i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.48 - 5.55i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (1.30 - 2.25i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.67 + 5.00i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.32 - 1.42i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 + (-14.8 + 3.98i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-13.1 + 7.57i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.42 + 9.42i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.52 - 9.57i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.48 + 2.48i)T - 97iT^{2} \) |
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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
−11.05575511957968531206393020761, −10.43580839577217731755774821947, −9.323526545915361967910334947569, −8.223586889173187438833620564824, −7.35210356486107514705700302078, −6.79677532231762024562820685666, −5.71999514154193588142192759124, −4.69153559607178379340219796102, −3.49629037629950585440529224969, −1.69600912448547546309212986254,
0.827497635282749631470069341720, 2.34906500676949441231517626274, 3.37343858910376367475928723635, 5.17327193501978900586295133642, 5.82868595169046041660178064874, 6.74795500514971140678200523115, 8.026460479207935937297528311561, 8.887468362682738287576488488886, 9.939980945069811217981048231740, 10.82255320385257032735729869377