L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)14-s + (0.499 + 0.866i)15-s + 16-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)20-s + (−0.499 − 0.866i)21-s + ⋯ |
L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)14-s + (0.499 + 0.866i)15-s + 16-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)20-s + (−0.499 − 0.866i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7609237508\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7609237508\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.733517709600166635968790034880, −8.595604449712224635771108269590, −8.115233629884421508509022543006, −7.32288600453071522221585173184, −6.83026847723848053162463416563, −6.08140068396568642652600271847, −4.35270252152875581853806743661, −3.14053318006425320907809437051, −2.33821250662494675158540583083, −0.918707549892505776501049893952,
1.58125098722236226816032540179, 2.82687086536661439279922764917, 3.86255355668347233751300706722, 5.13723730185795925571389087856, 5.62584183176806742642517594738, 7.18672108008534280345880671456, 7.906275736739085378071031982733, 8.803774202252009732078253178272, 8.949693717962469567784953589874, 9.792943128739386561937933465380