L(s) = 1 | + (0.5 + 0.363i)5-s + (−0.190 + 0.587i)7-s + (0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + (0.690 − 0.951i)17-s + (0.309 + 0.951i)19-s − 1.17i·23-s + (−0.190 − 0.587i)25-s + (−0.309 + 0.224i)35-s + 1.61·43-s + 0.618·45-s + (−1.11 + 0.363i)47-s + (0.5 + 0.363i)49-s + (0.190 − 0.587i)55-s + (−0.690 + 0.951i)61-s + ⋯ |
L(s) = 1 | + (0.5 + 0.363i)5-s + (−0.190 + 0.587i)7-s + (0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + (0.690 − 0.951i)17-s + (0.309 + 0.951i)19-s − 1.17i·23-s + (−0.190 − 0.587i)25-s + (−0.309 + 0.224i)35-s + 1.61·43-s + 0.618·45-s + (−1.11 + 0.363i)47-s + (0.5 + 0.363i)49-s + (0.190 − 0.587i)55-s + (−0.690 + 0.951i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.456182031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456182031\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + 1.17iT - T^{2} \) |
| 29 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61T + T^{2} \) |
| 47 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−8.882020133466705844281154467880, −7.999881242392125560902026065797, −7.30770359615706769292773794868, −6.30547924667125235796086804949, −5.97603350547300033637005112840, −5.06037803233047687876013795735, −4.04219925040875579592204117063, −3.10891848556921176862480591615, −2.38211290913058535885920937498, −1.02047796032043042687454904037,
1.32134284709391529844119437901, 2.09744179524927712456604332889, 3.38043888959729915668037197826, 4.26813986721624516867527644553, 5.02044540650871994146841926078, 5.69196380740912817677448220604, 6.72513380474133564691401112198, 7.46657531543556181028425075230, 7.82515447144008978870911094846, 8.984318338853886291624747003802