L(s) = 1 | − 0.963·3-s + 3.73·5-s + 0.145i·7-s − 2.07·9-s − i·11-s − 5.37i·13-s − 3.59·15-s + 1.20·17-s + (−3.41 + 2.70i)19-s − 0.140i·21-s + 6.87i·23-s + 8.93·25-s + 4.88·27-s + 1.91i·29-s + 6.32·31-s + ⋯ |
L(s) = 1 | − 0.556·3-s + 1.66·5-s + 0.0549i·7-s − 0.690·9-s − 0.301i·11-s − 1.48i·13-s − 0.928·15-s + 0.292·17-s + (−0.784 + 0.620i)19-s − 0.0305i·21-s + 1.43i·23-s + 1.78·25-s + 0.940·27-s + 0.355i·29-s + 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.857341912\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.857341912\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 19 | \( 1 + (3.41 - 2.70i)T \) |
good | 3 | \( 1 + 0.963T + 3T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 0.145iT - 7T^{2} \) |
| 13 | \( 1 + 5.37iT - 13T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 23 | \( 1 - 6.87iT - 23T^{2} \) |
| 29 | \( 1 - 1.91iT - 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 + 8.26iT - 37T^{2} \) |
| 41 | \( 1 + 7.55iT - 41T^{2} \) |
| 43 | \( 1 + 3.30iT - 43T^{2} \) |
| 47 | \( 1 + 9.21iT - 47T^{2} \) |
| 53 | \( 1 + 0.682iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.20T + 61T^{2} \) |
| 67 | \( 1 - 8.98T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 9.94T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 1.69iT - 83T^{2} \) |
| 89 | \( 1 + 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 8.95iT - 97T^{2} \) |
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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
−8.624217100196958928347955100053, −7.83081434571519213730370720080, −6.80244698257710532486594132480, −5.97135551171736979786145990533, −5.55318377393919562332534245160, −5.20204267902271360189382876685, −3.69824151326677589812338786700, −2.77088080394486724655493372566, −1.89404063673142779702344711324, −0.65879658267108859005373263081,
1.11394268225538689461265537594, 2.22945131258373530658076834732, 2.81577193877625352676440588836, 4.47010127068156115337557507412, 4.85863080938109743060445034626, 5.92059742622316416497656673533, 6.45093751567018250897301325584, 6.75499285279013093776171390766, 8.182625779613402099441450851339, 8.815130922858522005530391137240