L(s) = 1 | + (−0.642 + 1.26i)3-s + (0.951 + 0.309i)4-s + (−0.587 + 0.809i)5-s + (−0.587 − 0.809i)9-s + (−1 + i)12-s + (−0.642 − 1.26i)15-s + (0.809 + 0.587i)16-s + (−0.809 + 0.587i)20-s + (−1 − i)23-s + (−0.309 − 0.951i)25-s + (−0.309 − 0.951i)36-s + (0.642 + 1.26i)37-s + 1.00·45-s + (1.26 + 0.642i)47-s + (−1.26 + 0.642i)48-s + (0.587 − 0.809i)49-s + ⋯ |
L(s) = 1 | + (−0.642 + 1.26i)3-s + (0.951 + 0.309i)4-s + (−0.587 + 0.809i)5-s + (−0.587 − 0.809i)9-s + (−1 + i)12-s + (−0.642 − 1.26i)15-s + (0.809 + 0.587i)16-s + (−0.809 + 0.587i)20-s + (−1 − i)23-s + (−0.309 − 0.951i)25-s + (−0.309 − 0.951i)36-s + (0.642 + 1.26i)37-s + 1.00·45-s + (1.26 + 0.642i)47-s + (−1.26 + 0.642i)48-s + (0.587 − 0.809i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8159426242\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8159426242\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 3 | \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-1.39 - 0.221i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 - 2iT - T^{2} \) |
| 97 | \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \) |
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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
−10.96416955774581564923394949824, −10.50681323268341012871413717064, −9.806823773817225625560878366607, −8.459186562246442588866754211121, −7.53666071087529629579232398598, −6.56876056037991456570817729949, −5.78501667323199032752383553393, −4.45377581246930556150005202452, −3.64250115166442234441480454658, −2.51612685095028633135248256053,
1.08259507175776341662098159118, 2.24008365969667372946248075787, 3.94135349979347231119537022174, 5.48791725371265767989687517071, 6.00101410166442674505115318748, 7.22926784594741112986842755403, 7.52277158992579536781293853955, 8.606061921058946658136938767838, 9.811973448822862727553915900723, 10.92087499033879831073394419719