L(s) = 1 | + 5-s − i·9-s + (−1 − i)13-s + (1 − i)17-s + 25-s + 2i·29-s + (−1 + i)37-s − i·45-s + i·49-s + (−1 − i)53-s + 2·61-s + (−1 − i)65-s + (1 + i)73-s − 81-s + (1 − i)85-s + ⋯ |
L(s) = 1 | + 5-s − i·9-s + (−1 − i)13-s + (1 − i)17-s + 25-s + 2i·29-s + (−1 + i)37-s − i·45-s + i·49-s + (−1 − i)53-s + 2·61-s + (−1 − i)65-s + (1 + i)73-s − 81-s + (1 − i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.234704274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.234704274\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−9.799743619951564699777668367968, −9.186141973908284260333454055670, −8.248164377289671937300698850356, −7.17438654156441139716822791752, −6.57314167631080201188453586388, −5.41866129202010967020773057736, −5.05250576282773688095770727869, −3.43923422826357538745900098234, −2.70701320733826131018143723953, −1.20109791377981409238931281914,
1.77841542129944846033303856091, 2.48842071831408910394253778505, 3.96212254881160499698511916202, 4.99125058016544368812370535309, 5.70376606193782252549884416415, 6.59235801628521209496711360420, 7.52536394127897905866697190257, 8.291711220449639671757808819552, 9.291224714006104767552104887300, 9.964832605681071972461891305552