| L(s) = 1 | + (0.0871 + 0.996i)2-s + (−0.913 − 1.47i)3-s + (−0.984 + 0.173i)4-s + (1.19 − 1.88i)5-s + (1.38 − 1.03i)6-s + (−2.41 − 0.645i)7-s + (−0.258 − 0.965i)8-s + (−1.32 + 2.68i)9-s + (1.98 + 1.03i)10-s + (−3.63 − 2.09i)11-s + (1.15 + 1.29i)12-s + (0.156 + 0.336i)13-s + (0.433 − 2.45i)14-s + (−3.87 − 0.0395i)15-s + (0.939 − 0.342i)16-s + (0.633 + 7.24i)17-s + ⋯ |
| L(s) = 1 | + (0.0616 + 0.704i)2-s + (−0.527 − 0.849i)3-s + (−0.492 + 0.0868i)4-s + (0.536 − 0.844i)5-s + (0.565 − 0.424i)6-s + (−0.911 − 0.244i)7-s + (−0.0915 − 0.341i)8-s + (−0.443 + 0.896i)9-s + (0.627 + 0.325i)10-s + (−1.09 − 0.632i)11-s + (0.333 + 0.372i)12-s + (0.0435 + 0.0933i)13-s + (0.115 − 0.656i)14-s + (−0.999 − 0.0102i)15-s + (0.234 − 0.0855i)16-s + (0.153 + 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00956248 - 0.165839i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00956248 - 0.165839i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0871 - 0.996i)T \) |
| 3 | \( 1 + (0.913 + 1.47i)T \) |
| 5 | \( 1 + (-1.19 + 1.88i)T \) |
| 19 | \( 1 + (4.01 - 1.70i)T \) |
| good | 7 | \( 1 + (2.41 + 0.645i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.63 + 2.09i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.156 - 0.336i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.633 - 7.24i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-1.85 - 1.30i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (7.94 - 6.66i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.31 + 5.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.34 + 1.34i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.365 + 1.00i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-5.76 + 4.03i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (9.89 + 0.865i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (1.26 + 0.883i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (5.63 + 4.73i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.67 + 9.51i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.228 + 2.61i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.22 - 0.744i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (7.78 + 3.62i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (-2.04 - 5.60i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-14.4 - 3.86i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (7.66 + 2.78i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-14.7 + 1.29i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.35610204042974554151308028363, −9.244719823452165825023021353440, −8.323759363543992128822245273165, −7.65543068585927147584099300563, −6.43230137865519618090314446155, −5.89262452068970187289674630766, −5.11900967674542023502107888520, −3.65355634486455585786675096566, −1.85463003863750710147750309448, −0.092111675283706811364566476448,
2.51848204981134023417986077907, 3.21961351289102790696198294670, 4.59790771666247304129948108953, 5.50151522024388012571524950937, 6.43947194811648432365524855268, 7.49241375819285183783503867222, 9.130675415480320186613928454797, 9.625359661054195341857984782136, 10.32430435428986666457196647215, 10.99404065896910205951974500909
