L(s) = 1 | + (1.67 + 0.965i)5-s + (−0.866 + 1.5i)13-s − 0.517i·17-s + (1.36 + 2.36i)25-s + (−0.448 + 0.258i)29-s + 37-s + (−1.22 − 0.707i)41-s + (0.5 − 0.866i)49-s − 1.41i·53-s + (0.5 + 0.866i)61-s + (−2.89 + 1.67i)65-s − 1.73·73-s + (0.499 − 0.866i)85-s − 1.93i·89-s + (1.22 − 0.707i)101-s + ⋯ |
L(s) = 1 | + (1.67 + 0.965i)5-s + (−0.866 + 1.5i)13-s − 0.517i·17-s + (1.36 + 2.36i)25-s + (−0.448 + 0.258i)29-s + 37-s + (−1.22 − 0.707i)41-s + (0.5 − 0.866i)49-s − 1.41i·53-s + (0.5 + 0.866i)61-s + (−2.89 + 1.67i)65-s − 1.73·73-s + (0.499 − 0.866i)85-s − 1.93i·89-s + (1.22 − 0.707i)101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.562369421\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.562369421\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 0.517iT - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.73T + T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 1.93iT - T^{2} \) |
| 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
−9.311997331130611806093866607675, −8.689371349192509916577877084368, −7.23002078781037903787834094009, −6.97935415248565338126010331714, −6.14446696228811313351898518521, −5.41701360641018319202953264942, −4.57082881717944504809096068991, −3.32155480308091806378386470994, −2.34558287810798183613684044533, −1.77178972164937112198459829017,
1.07218889833102105782213225997, 2.15341817469358275112318333243, 2.99786747201130203958712565753, 4.42630903189463552905820339022, 5.18961229231676812038582420873, 5.76584602048227487502722119520, 6.38624414490167699246640966297, 7.55504310026101643145430079824, 8.281534290972579413992206888999, 9.045263785252909376297345595343