L(s) = 1 | + (−0.587 − 0.190i)2-s + (−0.5 − 0.363i)4-s + (−0.951 + 0.309i)5-s + 0.618·7-s + (0.587 + 0.809i)8-s + 0.618·10-s + (0.951 + 0.309i)11-s + (−0.363 − 0.118i)14-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.587 + 0.190i)20-s + (−0.5 − 0.363i)22-s + (0.951 + 0.309i)23-s + (0.809 − 0.587i)25-s + (−0.309 − 0.224i)28-s + (0.951 − 1.30i)29-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.190i)2-s + (−0.5 − 0.363i)4-s + (−0.951 + 0.309i)5-s + 0.618·7-s + (0.587 + 0.809i)8-s + 0.618·10-s + (0.951 + 0.309i)11-s + (−0.363 − 0.118i)14-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.587 + 0.190i)20-s + (−0.5 − 0.363i)22-s + (0.951 + 0.309i)23-s + (0.809 − 0.587i)25-s + (−0.309 − 0.224i)28-s + (0.951 − 1.30i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5973092387\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5973092387\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
good | 2 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
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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
−10.71359555361942543207152476033, −9.633688827441210013404707014910, −9.052664599804769616819463547555, −8.112848376971759624915034676654, −7.40276854027150644464780329706, −6.35889741769014805492783212420, −4.84875106486348980955698254589, −4.39500173298016430344166545065, −2.86786606876247493927832887568, −1.15042548397837421120614910556,
1.21768196221950467882785111316, 3.40912325661779116458477850600, 4.21811413386085642503610431240, 5.12099084751868884820890713739, 6.64179760660620429496195091186, 7.47704953998446398537509451372, 8.354216706705459792519126730813, 8.768685106147639149243110641889, 9.656099006697641898073246831803, 10.83983073379639820818263826831