L(s) = 1 | + (−0.5 − 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s − 1.99·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.999 − 1.73i)10-s + (−0.5 + 0.866i)11-s + (0.999 + 1.73i)12-s + 2·13-s + 3.99·15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (1 + 1.73i)19-s − 1.99·20-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s − 0.816·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−0.150 + 0.261i)11-s + (0.288 + 0.499i)12-s + 0.554·13-s + 1.03·15-s + (−0.125 − 0.216i)16-s + (−0.117 + 0.204i)18-s + (0.229 + 0.397i)19-s − 0.447·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.825907429\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.825907429\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 18T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.878480197869522918873785902868, −8.833126940553592347636614710483, −8.185673751954384129227300022030, −7.29904018668731221920527153075, −6.70386142176735825779759230728, −5.63999097904389062575008372897, −4.21956228880155036300121981627, −2.98330363049331644095271697287, −2.30660920713042264604131737042, −1.21874606892508270594140318378,
1.10051110655016347381152921607, 2.79555753753060316965791167108, 4.02676480271910917857165975411, 4.80590847402103469607571089897, 5.68212035210045430791930913610, 6.58148929550446217701169107224, 7.78695518664863755018312269860, 8.528670673808767553411287838081, 9.277313174607398447291313851018, 9.569132257440736583791078470820