| 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 + (−0.408 + 0.0869i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (0.104 + 0.994i)10-s + (−1.06 − 1.18i)11-s + (−0.913 − 0.406i)12-s + (−5.43 + 2.41i)13-s + (0.279 − 0.310i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−2.85 + 3.16i)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 + (−0.154 + 0.0328i)7-s + (0.109 + 0.336i)8-s + (−0.326 − 0.0693i)9-s + (0.0330 + 0.314i)10-s + (−0.322 − 0.357i)11-s + (−0.263 − 0.117i)12-s + (−1.50 + 0.671i)13-s + (0.0747 − 0.0830i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.691 + 0.768i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0199348 + 0.0931137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0199348 + 0.0931137i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.561 - 5.53i)T \) |
| good | 7 | \( 1 + (0.408 - 0.0869i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (1.06 + 1.18i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (5.43 - 2.41i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (2.85 - 3.16i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.74 + 0.777i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.564 + 1.73i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (1.38 - 1.00i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-5.25 - 9.09i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.807 + 7.68i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-0.149 - 0.0666i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (2.85 + 2.07i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.75 + 0.373i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.490 + 4.66i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 + (5.59 - 9.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.31 - 1.55i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (0.618 + 0.686i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (7.00 - 7.78i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.138 + 1.31i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-1.71 + 5.26i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0901 - 0.277i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.24774219642309725567313948918, −9.462827569570916781680320540371, −8.651955201757650876380594803226, −8.011462398605469636534317855068, −6.98683690056573854268756091159, −6.42888530433296862095259277750, −5.34968862248739040376479375259, −4.42838594351117995738589841399, −2.72666174832766637764434283343, −1.66500658819474008650629694797,
0.04957703186415566341201908032, 2.22335613455923743780295569349, 2.96448613894639466814874919646, 4.26006654697124951799974830207, 5.18032642790565807562669607007, 6.34029287490553680637685828185, 7.44169707368693843059927980174, 7.971911283429868458861232050174, 9.332141588024015835526389352657, 9.618040449585784221130234380870
