L(s) = 1 | + 7-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)19-s + 25-s + (0.5 − 0.866i)31-s + (−1 + 1.73i)37-s + (0.5 − 0.866i)43-s + 49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (−1 − 1.73i)91-s + (0.5 − 0.866i)97-s + 2·103-s + ⋯ |
L(s) = 1 | + 7-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)19-s + 25-s + (0.5 − 0.866i)31-s + (−1 + 1.73i)37-s + (0.5 − 0.866i)43-s + 49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 + 0.866i)73-s + (−1 − 1.73i)79-s + (−1 − 1.73i)91-s + (0.5 − 0.866i)97-s + 2·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.277778509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277778509\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.965197137639307565490345864460, −8.359206202659616191092218338091, −7.57190484254547160774300182641, −7.04600569080584849253110280983, −5.81265083752014871343383155148, −5.10522681633471481915527360708, −4.53575872427349756876920266939, −3.18668971611496039336755154062, −2.42912079975785856482785934256, −0.982146521603076594537823915266,
1.49557311344948550455527650027, 2.35274301910678028761151887825, 3.65645765918656192194199195432, 4.62358964249994656542382993997, 5.11748946445603479251160522865, 6.22124977003612519667269099114, 7.10889829460470406674706521038, 7.64176372835243648768349244139, 8.611689903288101238030312326419, 9.188754919180011856069537541088