L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.991 − 0.130i)5-s + (0.707 + 0.707i)8-s + (0.866 + 0.5i)9-s + (0.130 + 0.991i)10-s + (0.541 − 0.541i)13-s + (0.500 − 0.866i)16-s + (0.478 − 1.78i)17-s + (0.258 − 0.965i)18-s + (0.923 − 0.382i)20-s + (0.965 + 0.258i)25-s + (−0.662 − 0.382i)26-s − 1.41i·29-s + (−0.965 − 0.258i)32-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.991 − 0.130i)5-s + (0.707 + 0.707i)8-s + (0.866 + 0.5i)9-s + (0.130 + 0.991i)10-s + (0.541 − 0.541i)13-s + (0.500 − 0.866i)16-s + (0.478 − 1.78i)17-s + (0.258 − 0.965i)18-s + (0.923 − 0.382i)20-s + (0.965 + 0.258i)25-s + (−0.662 − 0.382i)26-s − 1.41i·29-s + (−0.965 − 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0287 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0287 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7455866289\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7455866289\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 + (0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 17 | \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.84iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
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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
−9.978222414499948369425590883124, −9.427425915168440542835965090405, −8.231586789619067901601384963127, −7.82344513262121295631256847138, −6.90406091070409591491873705145, −5.26598369052086314353096359777, −4.50541740752553662359385045715, −3.60380279595994881584029516871, −2.57688574310364587128843940915, −0.966980744804699731186136184365,
1.37713985780495308752362656855, 3.69934576836209754967876138625, 4.10286042903481490293795281353, 5.32430360596418271828132011195, 6.42207768226677847992258190346, 7.02117417152306229068791135966, 7.86510711626719839974681454150, 8.604930416452370074030050195830, 9.310802037866068194401970868840, 10.43508583033447965544315011821