L(s) = 1 | + (0.243 + 0.0651i)3-s + (0.356 − 2.20i)5-s + (−4.30 − 2.48i)7-s + (−2.54 − 1.46i)9-s + (1.07 − 4.01i)11-s + (3.20 + 1.65i)13-s + (0.230 − 0.513i)15-s + (1.65 + 6.16i)17-s + (1.01 − 0.270i)19-s + (−0.885 − 0.885i)21-s + (0.195 − 0.730i)23-s + (−4.74 − 1.57i)25-s + (−1.05 − 1.05i)27-s + (6.59 − 3.80i)29-s + (2.17 − 2.17i)31-s + ⋯ |
L(s) = 1 | + (0.140 + 0.0375i)3-s + (0.159 − 0.987i)5-s + (−1.62 − 0.940i)7-s + (−0.847 − 0.489i)9-s + (0.324 − 1.21i)11-s + (0.888 + 0.459i)13-s + (0.0594 − 0.132i)15-s + (0.400 + 1.49i)17-s + (0.231 − 0.0621i)19-s + (−0.193 − 0.193i)21-s + (0.0408 − 0.152i)23-s + (−0.949 − 0.314i)25-s + (−0.203 − 0.203i)27-s + (1.22 − 0.706i)29-s + (0.390 − 0.390i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.681873 - 0.760791i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.681873 - 0.760791i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.356 + 2.20i)T \) |
| 13 | \( 1 + (-3.20 - 1.65i)T \) |
good | 3 | \( 1 + (-0.243 - 0.0651i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (4.30 + 2.48i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.07 + 4.01i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.65 - 6.16i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.01 + 0.270i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.195 + 0.730i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-6.59 + 3.80i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.17 + 2.17i)T - 31iT^{2} \) |
| 37 | \( 1 + (-5.36 + 3.09i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.57 + 0.689i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (2.90 - 0.779i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 - 1.62iT - 47T^{2} \) |
| 53 | \( 1 + (5.32 - 5.32i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.52 - 9.43i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.39 + 9.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.04 + 5.27i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.15 + 4.31i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 5.98T + 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 - 8.32iT - 83T^{2} \) |
| 89 | \( 1 + (1.99 + 0.534i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.17 + 7.23i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.83542329905021458427384802487, −10.71656845035548446747980820279, −9.696126572864970307443332021679, −8.874427877635879155317485681321, −8.090837725375692807618764593413, −6.29010239830171365546968031218, −6.05167127888850021965667272658, −4.08602240146916059361175829464, −3.27460503764728039820001165498, −0.799381073872753169740330827746,
2.61086426622704848495500029336, 3.29002455466659359925234096891, 5.28283128841890668683678532959, 6.35052635060878790420697593036, 7.08852909058562221221086818038, 8.461943473617902563278296611590, 9.579574071770131106338299689101, 10.10741734472681114414236586166, 11.39608422686757213911566796801, 12.17614362880007508657163141276