| L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.913 − 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.773 + 0.858i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (3.11 + 0.663i)11-s + (0.104 − 0.994i)12-s + (−0.111 − 1.06i)13-s + (1.13 − 0.240i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (−4.00 + 0.851i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.527 − 0.234i)3-s + (0.154 + 0.475i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.292 + 0.324i)7-s + (0.109 − 0.336i)8-s + (0.223 + 0.247i)9-s + (0.288 − 0.128i)10-s + (0.940 + 0.199i)11-s + (0.0301 − 0.287i)12-s + (−0.0309 − 0.294i)13-s + (0.302 − 0.0642i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.972 + 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.267816 + 0.347349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267816 + 0.347349i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (5.09 - 2.23i)T \) |
| good | 7 | \( 1 + (0.773 - 0.858i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.11 - 0.663i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.111 + 1.06i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (4.00 - 0.851i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.544 + 5.17i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.83 - 5.65i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.42 - 3.21i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (4.67 + 8.09i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.41 - 2.85i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.955 - 9.08i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (7.21 - 5.24i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.11 - 9.01i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (8.61 + 3.83i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + (-0.597 + 1.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.95 + 4.38i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (10.8 + 2.31i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-6.41 + 1.36i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (15.0 - 6.67i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-5.28 - 16.2i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.52 - 7.77i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.43419802571119986822621475123, −9.391327076849017032491268926907, −8.897801225640561020698611492368, −7.73720583733499097207715531697, −6.91367547007583107147074394524, −6.29137130824830816337013979462, −5.04197884109551878509493830152, −3.88483213302060061528173122813, −2.77271216412497433096163479994, −1.46046116070495221606818497128,
0.27574583759298956658740386828, 1.79577388043798802366623055452, 3.67175663928836707288165712452, 4.53245603220601806261194484928, 5.61274327903470006435780379348, 6.57643623523807731145675685531, 7.06041707436344357754449137939, 8.436926665265774410949662812393, 8.779729441371075265785866668166, 10.02694518271688097554689705865
