L(s) = 1 | + (−0.587 − 0.809i)2-s + (0.587 − 0.809i)3-s + (−0.309 + 0.951i)4-s + (−1.61 − 1.54i)5-s − 6-s + (−3.49 − 1.13i)7-s + (0.951 − 0.309i)8-s + (−0.309 − 0.951i)9-s + (−0.297 + 2.21i)10-s + (−0.348 + 1.07i)11-s + (0.587 + 0.809i)12-s + (3.06 − 4.21i)13-s + (1.13 + 3.49i)14-s + (−2.19 + 0.401i)15-s + (−0.809 − 0.587i)16-s + (−3.06 + 0.997i)17-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.339 − 0.467i)3-s + (−0.154 + 0.475i)4-s + (−0.723 − 0.690i)5-s − 0.408·6-s + (−1.31 − 0.428i)7-s + (0.336 − 0.109i)8-s + (−0.103 − 0.317i)9-s + (−0.0940 + 0.700i)10-s + (−0.105 + 0.323i)11-s + (0.169 + 0.233i)12-s + (0.850 − 1.16i)13-s + (0.303 + 0.933i)14-s + (−0.567 + 0.103i)15-s + (−0.202 − 0.146i)16-s + (−0.744 + 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0661743 + 0.0758470i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0661743 + 0.0758470i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (1.61 + 1.54i)T \) |
| 31 | \( 1 + (-5.55 - 0.331i)T \) |
good | 7 | \( 1 + (3.49 + 1.13i)T + (5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.348 - 1.07i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-3.06 + 4.21i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.06 - 0.997i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.27 - 0.923i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.25 + 0.408i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.55 - 4.03i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 - 9.34iT - 37T^{2} \) |
| 41 | \( 1 + (8.91 - 6.47i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (0.705 + 0.970i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (1.10 - 1.52i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.40 + 0.457i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (7.57 + 5.50i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 - 0.883T + 61T^{2} \) |
| 67 | \( 1 + 4.69iT - 67T^{2} \) |
| 71 | \( 1 + (1.31 + 4.06i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.09 - 0.357i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.295 - 0.908i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.67 + 3.67i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.382 - 1.17i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (9.54 + 3.10i)T + (78.4 + 57.0i)T^{2} \) |
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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
−9.462039181877196913271615424619, −8.543389184369334405598797004913, −8.057260486922888535040211351420, −7.04154438598323656734373427339, −6.22386500101188182043619919614, −4.80267799368496085909342764452, −3.63220714880503811606816248816, −3.04515586793649564210637367258, −1.36857365010203890206368124043, −0.05391273898930158867887671363,
2.39273408067590305859004630740, 3.53986089173660062218028638217, 4.30590429726432337499060571418, 5.76916722193669830786284472385, 6.58188819478638322945612256656, 7.16255603850159433053960669003, 8.328190781201050995045996209645, 9.002721688989082723711204324760, 9.608633057697728781384613416997, 10.57915551823001739131003098660