| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.514 + 3.57i)5-s + (−0.841 − 0.540i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (−1.50 − 3.28i)10-s + (2.13 + 0.627i)11-s + (0.959 + 0.281i)12-s + (0.498 + 1.09i)13-s + (0.142 − 0.989i)14-s + (−2.36 + 2.73i)15-s + (0.415 − 0.909i)16-s + (4.28 + 2.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (0.230 + 1.60i)5-s + (−0.343 − 0.220i)6-s + (−0.157 + 0.343i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (−0.474 − 1.03i)10-s + (0.644 + 0.189i)11-s + (0.276 + 0.0813i)12-s + (0.138 + 0.302i)13-s + (0.0380 − 0.264i)14-s + (−0.611 + 0.705i)15-s + (0.103 − 0.227i)16-s + (1.03 + 0.667i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.524616 + 1.25418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524616 + 1.25418i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.78 + 0.265i)T \) |
| good | 5 | \( 1 + (-0.514 - 3.57i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-2.13 - 0.627i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.498 - 1.09i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.28 - 2.75i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-5.04 + 3.24i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.27 - 3.39i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (7.20 - 8.31i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.51 + 10.5i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.327 + 2.27i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.20 + 4.85i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 4.54T + 47T^{2} \) |
| 53 | \( 1 + (2.98 - 6.52i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.40 + 9.65i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.25 + 2.60i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (14.2 - 4.19i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-6.40 + 1.88i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (1.52 - 0.983i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.228 - 0.499i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.32 + 9.20i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.80 - 7.84i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.0489 - 0.340i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.37737314539282100305816578489, −9.457969281017911940099652277225, −8.892818892421923802020672597100, −7.71044370450493761136346302089, −7.07007402172393313290867327563, −6.27101289313918442932684789628, −5.35628615348667370451338880749, −3.72256559358867157527883668806, −2.99591902105924999262444127523, −1.82119459188818861838201325389,
0.810406353896030340965750477997, 1.59429688059629383105077125811, 3.15929371438561646143203615636, 4.25173883236590050187849997546, 5.47280040951003635774356568237, 6.29526023339081622487655394695, 7.67052012661674781767379164831, 8.001362317999179842358942089649, 8.926182216574077418153981434954, 9.661336125767059984838280518204
