| L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.142 + 0.989i)6-s + (−3.76 + 1.10i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (0.883 + 1.01i)11-s + (0.654 + 0.755i)12-s + (1.39 + 0.408i)13-s + (−1.63 + 3.57i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (−0.936 + 6.51i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.185 − 0.406i)5-s + (−0.0580 + 0.404i)6-s + (−1.42 + 0.418i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (0.266 + 0.307i)11-s + (0.189 + 0.218i)12-s + (0.385 + 0.113i)13-s + (−0.435 + 0.954i)14-s + (0.217 + 0.139i)15-s + (−0.239 + 0.0704i)16-s + (−0.227 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0407 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0407 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.447517 + 0.466141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.447517 + 0.466141i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (4.52 - 1.59i)T \) |
| good | 7 | \( 1 + (3.76 - 1.10i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.883 - 1.01i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.39 - 0.408i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.936 - 6.51i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.354 - 2.46i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.10 - 7.70i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.10 - 1.35i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.371 + 0.814i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (4.24 + 9.30i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.59 - 4.23i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + (-12.1 + 3.56i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (8.65 + 2.54i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.74 - 4.33i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (8.07 - 9.31i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.95 + 8.02i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.588 + 4.09i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-8.99 - 2.64i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.61 + 5.72i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (13.5 - 8.69i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.33 - 5.11i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.58173592319994296159155940852, −10.03275572751026229108689243088, −9.189644149861746467295754192836, −8.335827074832436635804297387556, −6.77993980476148125802366449826, −6.11237840580633491936694892638, −5.26495554082953317798159179549, −3.98417248822314859507880955128, −3.39704919646725045761434798434, −1.70600338560912013334500168446,
0.30273587815566595988893483838, 2.72064379028996250433707113924, 3.69375175048975303071241888672, 4.81577002808505678299171648521, 6.07867896750176357280899262975, 6.57386910894784214653741968780, 7.29394313894555280002196154427, 8.294835837091243460615760654128, 9.523872396770458066340506689633, 10.14032788920148094280762682512
