L(s) = 1 | + (0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s + (−0.999 + 1.73i)6-s − 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 + 2.59i)10-s + 11-s − 1.99·12-s + (2 − 3.46i)13-s + (−0.5 − 0.866i)14-s + (−3 + 5.19i)15-s + (−0.5 − 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.408 + 0.707i)6-s − 0.377·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.474 + 0.821i)10-s + 0.301·11-s − 0.577·12-s + (0.554 − 0.960i)13-s + (−0.133 − 0.231i)14-s + (−0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.855300 + 1.88627i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.855300 + 1.88627i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + (3.5 + 2.59i)T \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.25664942521452835462549044770, −10.43676169019491811916616801470, −9.736260757540450817237931022684, −8.904507211767582950778476744323, −7.84382001674572726484910320123, −6.59567123484551413667535859853, −6.03595373463731358190879665573, −4.64185985594449241551876473203, −3.51764230267867490351764021950, −2.72105439189372660242162823600,
1.33074000930363262732815613750, 2.15524718829239426754884768100, 3.75231918841972684403261549152, 4.95985926861868640547139133272, 6.11317790269323866411209143952, 7.03620393887706903165464499772, 8.467031651044092260009441276899, 8.924686139704274987518687740663, 9.877362423057999865143123180099, 10.98353865382120255724170948324