| L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.104 + 0.994i)3-s + (0.309 + 0.951i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (0.272 + 0.0578i)7-s + (0.309 − 0.951i)8-s + (−0.978 + 0.207i)9-s + (−0.104 + 0.994i)10-s + (−0.538 + 0.597i)11-s + (−0.913 + 0.406i)12-s + (−3.82 − 1.70i)13-s + (−0.186 − 0.206i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (4.46 + 4.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.0603 + 0.574i)3-s + (0.154 + 0.475i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + (0.102 + 0.0218i)7-s + (0.109 − 0.336i)8-s + (−0.326 + 0.0693i)9-s + (−0.0330 + 0.314i)10-s + (−0.162 + 0.180i)11-s + (−0.263 + 0.117i)12-s + (−1.06 − 0.472i)13-s + (−0.0497 − 0.0552i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (1.08 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.178564 + 0.399379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.178564 + 0.399379i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.44 - 1.14i)T \) |
| good | 7 | \( 1 + (-0.272 - 0.0578i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (0.538 - 0.597i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (3.82 + 1.70i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-4.46 - 4.96i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (5.23 - 2.33i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.320 + 0.985i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 0.889i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (4.82 - 8.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.311 - 2.96i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-1.48 + 0.661i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-9.06 + 6.58i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (11.0 - 2.35i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.504 - 4.79i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + (-5.33 - 9.23i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.9 - 2.32i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (2.05 - 2.28i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.656 + 0.729i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.402 - 3.82i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (2.74 + 8.44i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.70 - 11.4i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.31045120791401051374772740327, −9.719893176198313714145397560210, −8.660382139631149662279176954101, −8.148256667986969840443546845825, −7.26496775940115460144149840351, −5.99357951914733017469974488449, −4.97379328371698807336216972382, −4.02178459880526824193219418278, −2.99892225002277851139141196097, −1.65463108834686069152886434776,
0.23702070622272014086283723928, 1.96780338482640116090593851311, 3.05395124869266515744770056282, 4.57841960919848820703814704334, 5.59375881419070038783899797322, 6.58420104518981393471886547212, 7.41407140391570574155965980110, 7.77177823427443990494195758422, 8.981465608920604440649189985560, 9.528746690489316692841365112436
