| L(s) = 1 | − i·3-s + (−1 − i)7-s − 9-s − i·13-s + (1 − i)19-s + (−1 + i)21-s + i·25-s + i·27-s + (−1 + i)31-s + (1 − i)37-s − 39-s + 2·43-s + i·49-s + (−1 − i)57-s + (1 + i)63-s + ⋯ |
| L(s) = 1 | − i·3-s + (−1 − i)7-s − 9-s − i·13-s + (1 − i)19-s + (−1 + i)21-s + i·25-s + i·27-s + (−1 + i)31-s + (1 − i)37-s − 39-s + 2·43-s + i·49-s + (−1 − i)57-s + (1 + i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7744806690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7744806690\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - 2T + T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 - i)T - iT^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 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
−10.71197930365671610953105604499, −9.631990022786358968870835853678, −8.841630836075293556161477061963, −7.41116142691185022162072803189, −7.35591247031106411132760338693, −6.17099332245191928173702122491, −5.26447252812542490775105176132, −3.66250277149528561884433183962, −2.74767561213197130896813141058, −0.919220630484842087631974528060,
2.43561245836311904086621093138, 3.50819527704168511020135109670, 4.51227218613201695542567394735, 5.74343930676503148968667070993, 6.25841457990201291275324435694, 7.67518730519710932554227260605, 8.790782976908530853575516345118, 9.459302909875637121571843867806, 9.944540973359718320332965974933, 11.05166707889653930175786089572
