L(s) = 1 | + (0.707 + 0.707i)3-s + (0.866 − 0.5i)5-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (0.965 + 0.258i)15-s + (−0.500 + 0.866i)21-s + (0.258 + 0.448i)23-s + (0.499 − 0.866i)25-s + (−0.707 + 0.707i)27-s − 1.73i·29-s + (0.707 + 0.707i)35-s + i·41-s + 1.93i·43-s + (0.500 + 0.866i)45-s + (−0.707 − 1.22i)47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (0.866 − 0.5i)5-s + (0.258 + 0.965i)7-s + 1.00i·9-s + (0.965 + 0.258i)15-s + (−0.500 + 0.866i)21-s + (0.258 + 0.448i)23-s + (0.499 − 0.866i)25-s + (−0.707 + 0.707i)27-s − 1.73i·29-s + (0.707 + 0.707i)35-s + i·41-s + 1.93i·43-s + (0.500 + 0.866i)45-s + (−0.707 − 1.22i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.936969251\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.936969251\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.258 - 0.965i)T \) |
good | 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - iT - T^{2} \) |
| 43 | \( 1 - 1.93iT - T^{2} \) |
| 47 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 1.93T + T^{2} \) |
| 89 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.019251853475402404479289638055, −8.250482650699014809978511718117, −7.82209657454645100709361537179, −6.47320596543721762544148666502, −5.80173019865100834177135592628, −5.00224437755434843561766955245, −4.46432520084043131354822745646, −3.26094377914878128619017677329, −2.44449803557646173338612020518, −1.64990733825982099400825637658,
1.18001691211894767998804880168, 2.05588211203471848073006824844, 3.03193268549033210467863904678, 3.77116944340870452643935491305, 4.87514563473481600312627489616, 5.83186993850269227713097328122, 6.69867987506764143152044779472, 7.15212713073435000062166946640, 7.77802920864979150783865623364, 8.818052349504061120590683876352