L(s) = 1 | + 1.41i·7-s + (1 + i)13-s − 6·17-s + (−4.24 − 4.24i)19-s − 8.48i·23-s + 5i·25-s + (−6 − 6i)29-s + 1.41·31-s + (5 − 5i)37-s + 6i·41-s + (4.24 − 4.24i)43-s − 8.48·47-s + 5·49-s + (6 − 6i)53-s + (5 + 5i)61-s + ⋯ |
L(s) = 1 | + 0.534i·7-s + (0.277 + 0.277i)13-s − 1.45·17-s + (−0.973 − 0.973i)19-s − 1.76i·23-s + i·25-s + (−1.11 − 1.11i)29-s + 0.254·31-s + (0.821 − 0.821i)37-s + 0.937i·41-s + (0.646 − 0.646i)43-s − 1.23·47-s + 0.714·49-s + (0.824 − 0.824i)53-s + (0.640 + 0.640i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8244834906\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8244834906\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (4.24 + 4.24i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (6 + 6i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 + 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + (-6 + 6i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (-5 - 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.82 + 2.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.845914283096735284626699754425, −8.163451320712628939103101405889, −7.07897673077629750437062174058, −6.46714280787397210439488723293, −5.71073416294572733335386709075, −4.61527511519245317811381811738, −4.07282440069341845717894324556, −2.67615584958346066760209047680, −2.03104027374006195044833112275, −0.27587155425655764035961868494,
1.35511617079620690035520217638, 2.45647611717807422851071426984, 3.70027810948321612485881886747, 4.27062819085745809150463364375, 5.34444544885517623226941439621, 6.17597995783632056657631792267, 6.92571965918598632690027979412, 7.72794468623125258241545305254, 8.465640868610099452471933417458, 9.203497233206157049903494076163