L(s) = 1 | + (−0.222 − 0.974i)2-s + (0.777 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (0.777 − 0.974i)6-s + (−0.900 + 0.433i)7-s + (0.623 + 0.781i)8-s + (−0.123 + 0.541i)9-s + (0.623 − 0.781i)10-s + (−1.12 − 0.541i)12-s + (0.623 + 0.781i)14-s + (−0.277 + 1.21i)15-s + (0.623 − 0.781i)16-s + 0.554·18-s + (−0.900 − 0.433i)20-s + (−1.12 − 0.541i)21-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (0.777 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (0.777 − 0.974i)6-s + (−0.900 + 0.433i)7-s + (0.623 + 0.781i)8-s + (−0.123 + 0.541i)9-s + (0.623 − 0.781i)10-s + (−1.12 − 0.541i)12-s + (0.623 + 0.781i)14-s + (−0.277 + 1.21i)15-s + (0.623 − 0.781i)16-s + 0.554·18-s + (−0.900 − 0.433i)20-s + (−1.12 − 0.541i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.061666841\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.061666841\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.623 - 0.781i)T \) |
| 7 | \( 1 + (0.900 - 0.433i)T \) |
good | 3 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 11 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 13 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 59 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 61 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.03380604048002688758657321360, −9.767858854661615763153280302170, −8.949601549504144116403298968847, −8.256484572749237839531320928635, −6.94730067371980265512361355836, −5.88205513803267081192465004157, −4.72643870121780806606004447202, −3.54707731269229620502256117192, −3.09155685357771908976336185125, −2.08331626889840893945009286381,
1.09258956676738256225905884825, 2.50501789478811885944457297494, 4.00177702331363115374721798981, 5.05306926467694393959481903420, 6.29056688324322807609563577078, 6.60423957190962728041240067808, 7.78151067333041282700668848573, 8.242058348248770192427819961255, 9.085015440858174323495712224396, 9.811547757035767958786928400273