| L(s) = 1 | + (0.959 + 0.281i)2-s + (1.04 + 0.907i)3-s + (0.841 + 0.540i)4-s + (0.229 − 1.59i)5-s + (0.749 + 1.16i)6-s + (2.30 − 1.29i)7-s + (0.654 + 0.755i)8-s + (−0.153 − 1.06i)9-s + (0.668 − 1.46i)10-s + (0.829 + 2.82i)11-s + (0.390 + 1.33i)12-s + (−4.50 − 2.05i)13-s + (2.57 − 0.592i)14-s + (1.68 − 1.46i)15-s + (0.415 + 0.909i)16-s + (−5.01 + 3.22i)17-s + ⋯ |
| L(s) = 1 | + (0.678 + 0.199i)2-s + (0.604 + 0.524i)3-s + (0.420 + 0.270i)4-s + (0.102 − 0.712i)5-s + (0.305 + 0.476i)6-s + (0.871 − 0.489i)7-s + (0.231 + 0.267i)8-s + (−0.0511 − 0.355i)9-s + (0.211 − 0.463i)10-s + (0.249 + 0.851i)11-s + (0.112 + 0.383i)12-s + (−1.25 − 0.571i)13-s + (0.689 − 0.158i)14-s + (0.435 − 0.377i)15-s + (0.103 + 0.227i)16-s + (−1.21 + 0.782i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33791 + 0.394681i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33791 + 0.394681i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (-2.30 + 1.29i)T \) |
| 23 | \( 1 + (4.66 - 1.11i)T \) |
| good | 3 | \( 1 + (-1.04 - 0.907i)T + (0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.229 + 1.59i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.829 - 2.82i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (4.50 + 2.05i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.01 - 3.22i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.72 - 2.39i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.58 - 1.65i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.64 - 4.02i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (4.07 - 0.585i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-8.19 - 1.17i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (6.07 + 5.26i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 5.61iT - 47T^{2} \) |
| 53 | \( 1 + (2.84 - 1.30i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-9.55 - 4.36i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (3.34 + 3.86i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.61 + 5.50i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.06 - 0.899i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-7.93 + 12.3i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (12.5 + 5.73i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.949 - 6.60i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.94 + 2.24i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (2.63 - 18.3i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.99206532493830380955194596042, −10.70557802064098108484403199848, −9.775964600816095265011262588359, −8.825037703082456196607640132180, −7.85982702935851479893570264529, −6.90869366284260359424622951577, −5.36947714473176086655458612571, −4.56407386151063029487138861988, −3.64469602449200681590686671890, −1.94924993708892502968523953470,
2.10480400062882473313296123484, 2.81474752861498901130128774832, 4.47406316895663046050949780086, 5.52414408918033844805830002809, 6.83871922930473612054121462928, 7.57204693557476757832627984098, 8.675592034234681029324144812460, 9.709741176925305664603086658554, 11.20757416098949604576129163375, 11.34677704576069952948997457558
