| L(s) = 1 | + i·2-s − 4-s + (1 + i)5-s − i·8-s + i·9-s + (−1 + i)10-s + 16-s + 17-s − 18-s + (−1 − i)20-s + i·25-s + (1 + i)29-s + i·32-s + i·34-s − i·36-s + (−1 − i)37-s + ⋯ |
| L(s) = 1 | + i·2-s − 4-s + (1 + i)5-s − i·8-s + i·9-s + (−1 + i)10-s + 16-s + 17-s − 18-s + (−1 − i)20-s + i·25-s + (1 + i)29-s + i·32-s + i·34-s − i·36-s + (−1 − i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.358232211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358232211\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-1 - i)T + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-1 - i)T + iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + (1 - i)T - iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 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
−8.988317510774996668541736067437, −8.202334424155764257956765858584, −7.50575887639937940949158821573, −6.81343652147486035026172025044, −6.21055863605209663476274700992, −5.37516842637644558425263657366, −4.90901771012926610417118480275, −3.63283968892761488034441673367, −2.76639974321406096970467182040, −1.58963277607082267870183528224,
0.894378655439629435642484357070, 1.70607229695763933127210759749, 2.82000327023422200456686663783, 3.71589776636210439291140395299, 4.60127189344037187120258141629, 5.40497688836295314121031880538, 5.94295530227994839951083086231, 6.97185243947968735825637832093, 8.218461130640259915870701265046, 8.697110020316634563007092859195
