L(s) = 1 | + (0.951 − 0.309i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.412i)7-s + (−0.587 + 0.809i)9-s + (0.951 + 0.309i)11-s + (0.809 − 0.587i)16-s + (−0.0489 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.587 + 0.809i)20-s + (1.39 − 1.39i)23-s + (−0.809 + 0.587i)25-s + (−0.896 − 0.142i)28-s + (0.142 − 0.896i)35-s + (−0.309 + 0.951i)36-s + (−1.39 − 1.39i)43-s + 0.999·44-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.412i)7-s + (−0.587 + 0.809i)9-s + (0.951 + 0.309i)11-s + (0.809 − 0.587i)16-s + (−0.0489 + 0.309i)17-s + (−0.309 + 0.951i)19-s + (0.587 + 0.809i)20-s + (1.39 − 1.39i)23-s + (−0.809 + 0.587i)25-s + (−0.896 − 0.142i)28-s + (0.142 − 0.896i)35-s + (−0.309 + 0.951i)36-s + (−1.39 − 1.39i)43-s + 0.999·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.240136574\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.240136574\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 3 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \) |
| 23 | \( 1 + (-1.39 + 1.39i)T - iT^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (1.39 + 1.39i)T + iT^{2} \) |
| 47 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.951 - 0.309i)T^{2} \) |
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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
−10.40299027851685110339392460519, −9.616202831958590247556163256259, −8.486796404254573692869262747553, −7.43601106532249989247807844853, −6.62384146548403989371945491649, −6.31836935589035941410877751032, −5.18572136423179205982101265104, −3.70503937752948131578165567643, −2.81605443740388489398273946657, −1.81024450389923655527135178909,
1.32945827878924088114668928209, 2.82967988266207884503276162076, 3.56061288568117692154673103124, 4.95701527880539416270085085726, 6.05129504686815895134734286029, 6.49013462795956675808768915428, 7.46216259221192162768545942451, 8.686988729004464045132040470627, 9.132504659882844439741009379474, 9.808288829702025144776560349954