L(s) = 1 | + (−0.309 + 0.951i)3-s + (0.951 − 0.309i)4-s − i·5-s + (−0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + 0.999i·12-s + (0.951 + 0.309i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)20-s + (0.142 + 0.896i)23-s − 25-s + (0.809 − 0.587i)27-s + (−0.363 + 1.11i)31-s + (0.587 + 0.809i)33-s + (−0.951 − 0.309i)36-s + (−1.39 − 0.221i)37-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)3-s + (0.951 − 0.309i)4-s − i·5-s + (−0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + 0.999i·12-s + (0.951 + 0.309i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)20-s + (0.142 + 0.896i)23-s − 25-s + (0.809 − 0.587i)27-s + (−0.363 + 1.11i)31-s + (0.587 + 0.809i)33-s + (−0.951 − 0.309i)36-s + (−1.39 − 0.221i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.089126899\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.089126899\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (-0.587 + 0.809i)T \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−10.59227814340391312955607888100, −9.492347674451474681731865905667, −8.977913857058806899397862206976, −7.986207839475145601437741091499, −6.79453439821690544556875867796, −5.79427043309097799065621908848, −5.27048785127432422817149779983, −4.07655107467333665645779168471, −3.07231977782587371939538560948, −1.36128657962000550185012757146,
1.84040638763147932381634540696, 2.63800235887392112417624965074, 3.84313449862990803194203262643, 5.46218372585940788903651114760, 6.51603998813645552489762575762, 6.87448913179785316035369646957, 7.59810552999309300554720697261, 8.464077879772912950762184474772, 9.802765100732143452137575283824, 10.72773099085772642578918514390