| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.376 + 0.825i)3-s + (−0.142 + 0.989i)4-s + (0.959 + 0.281i)5-s + (0.870 − 0.255i)6-s + (−1.54 − 1.78i)7-s + (0.841 − 0.540i)8-s + (1.42 + 1.64i)9-s + (−0.415 − 0.909i)10-s + (−5.01 − 1.47i)11-s + (−0.763 − 0.490i)12-s + (−2.73 − 1.75i)13-s + (−0.336 + 2.33i)14-s + (−0.594 + 0.685i)15-s + (−0.959 − 0.281i)16-s + (−0.416 − 2.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.217 + 0.476i)3-s + (−0.0711 + 0.494i)4-s + (0.429 + 0.125i)5-s + (0.355 − 0.104i)6-s + (−0.585 − 0.675i)7-s + (0.297 − 0.191i)8-s + (0.475 + 0.548i)9-s + (−0.131 − 0.287i)10-s + (−1.51 − 0.444i)11-s + (−0.220 − 0.141i)12-s + (−0.759 − 0.488i)13-s + (−0.0898 + 0.625i)14-s + (−0.153 + 0.177i)15-s + (−0.239 − 0.0704i)16-s + (−0.100 − 0.702i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378687 - 0.582512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378687 - 0.582512i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (0.641 + 8.16i)T \) |
| good | 3 | \( 1 + (0.376 - 0.825i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (1.54 + 1.78i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (5.01 + 1.47i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (2.73 + 1.75i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.416 + 2.89i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-5.60 + 6.46i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.117 + 0.257i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 + (3.16 - 2.03i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + 0.927T + 37T^{2} \) |
| 41 | \( 1 + (0.611 + 4.25i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.32 + 9.19i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-2.15 + 4.72i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (-1.33 + 9.26i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (9.49 - 6.10i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.442 + 0.129i)T + (51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (0.903 - 6.28i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (9.00 - 2.64i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (9.38 + 6.03i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-1.17 - 0.343i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-6.64 - 14.5i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
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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
−10.29407772660519732264041215839, −9.722552255312659956727644543050, −8.716549454942791559561488450958, −7.47773769943081204364202849495, −7.06813830736159340601698889762, −5.37077187472731568564558303265, −4.82152969233274044314420848555, −3.30129589160286290387789475442, −2.44718215173799970499109399833, −0.43958061201360602653168802516,
1.55947083775077507901320493934, 2.89108949514690224976117957656, 4.61141098788217822809908761379, 5.71838413539240806513039836217, 6.26923301643430033441490880598, 7.38589057856609395613656470509, 7.950503848699889425942486489746, 9.175892987904525590304879320853, 9.842321297278672920839370099071, 10.38426799851167813328122215871
