L(s) = 1 | + i·5-s + (−0.866 − 0.5i)9-s + (−1 − 1.73i)13-s + (−1.5 − 0.866i)17-s − 25-s + (−0.366 − 0.366i)29-s + (0.866 − 0.5i)37-s + (−1.5 + 0.866i)41-s + (0.5 − 0.866i)45-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + (1.86 + 0.5i)61-s + (1.73 − i)65-s + (1 + i)73-s + (0.499 + 0.866i)81-s + ⋯ |
L(s) = 1 | + i·5-s + (−0.866 − 0.5i)9-s + (−1 − 1.73i)13-s + (−1.5 − 0.866i)17-s − 25-s + (−0.366 − 0.366i)29-s + (0.866 − 0.5i)37-s + (−1.5 + 0.866i)41-s + (0.5 − 0.866i)45-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)53-s + (1.86 + 0.5i)61-s + (1.73 − i)65-s + (1 + i)73-s + (0.499 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4548068330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4548068330\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−8.534212107925110184689276793737, −7.961164256469178390680005997379, −7.06689421597939313226189128738, −6.52551735859523822814952566151, −5.61661105426379618982760777954, −4.94193320433058939994816512731, −3.71534200538352410732018693719, −2.88273374432666067305233115769, −2.33442474896933231737728510500, −0.25572418747778225709090047297,
1.72991289785876279656360418573, 2.40232501132904778095962122592, 3.84076512655674532085631246564, 4.62427354638850148906312380742, 5.14171191864966072784230624461, 6.18588692174135551500947673952, 6.84504634075586837777814614243, 7.82647992333480412773770700716, 8.558803216402892824481653024456, 9.073316607538249896801854410719