| L(s) = 1 | + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.694 − 3.93i)5-s + (0.939 − 0.342i)6-s + (−1.5 + 2.59i)7-s + (−0.500 − 0.866i)8-s + (−1.53 − 1.28i)9-s + (−3.06 − 2.57i)10-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)12-s + (0.939 − 0.342i)13-s + (0.520 + 2.95i)14-s + (0.694 − 3.93i)15-s + (−0.939 − 0.342i)16-s + (2.29 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.310 − 1.76i)5-s + (0.383 − 0.139i)6-s + (−0.566 + 0.981i)7-s + (−0.176 − 0.306i)8-s + (−0.510 − 0.428i)9-s + (−0.968 − 0.813i)10-s + (−0.301 − 0.522i)11-s + (0.144 − 0.250i)12-s + (0.260 − 0.0948i)13-s + (0.139 + 0.789i)14-s + (0.179 − 1.01i)15-s + (−0.234 − 0.0855i)16-s + (0.557 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.668258 - 1.63482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.668258 - 1.63482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-0.939 - 0.342i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.694 + 3.93i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.939 + 0.342i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.29 + 1.92i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (3.83 + 3.21i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-7.51 - 2.73i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.694 - 3.93i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.12 - 5.14i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.173 - 0.984i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-11.4 + 9.64i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.347 + 1.96i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.29 - 1.92i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.347 - 1.96i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (8.45 + 3.07i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-9.39 - 3.42i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.53 - 1.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.674404510318233069820676051713, −9.343398917884293226738179506364, −8.537268569271844048811657920178, −7.84580885527893242171555643276, −5.98673998840125542507550149284, −5.58805224378258564511893935610, −4.46093337790663584358666792915, −3.49298788144690490746115928958, −2.42855879150439763908586196773, −0.70305837674325201585939812283,
2.37531334459146690581550722978, 3.30990014845757975455377241577, 3.96658836716027935821134805837, 5.50758334305204991516361997592, 6.58295368802421572583584146115, 7.22988652874108129194175611748, 7.71472511650960678108561021459, 8.803858251130659719446568091212, 10.24474442807204545453415454734, 10.55757419238276451117637426550
