| L(s) = 1 | + (−1.87 − 0.684i)2-s + (−6.73 + 2.45i)3-s + (3.06 + 2.57i)4-s + (−2.75 + 15.6i)5-s + 14.3·6-s + (5.01 − 28.4i)7-s + (−4.00 − 6.92i)8-s + (18.6 − 15.6i)9-s + (15.8 − 27.5i)10-s + (−0.433 − 0.750i)11-s + (−26.9 − 9.80i)12-s + (−52.8 − 44.3i)13-s + (−28.8 + 50.0i)14-s + (−19.7 − 112. i)15-s + (2.77 + 15.7i)16-s + (47.0 − 39.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−1.29 + 0.471i)3-s + (0.383 + 0.321i)4-s + (−0.246 + 1.39i)5-s + 0.975·6-s + (0.270 − 1.53i)7-s + (−0.176 − 0.306i)8-s + (0.690 − 0.579i)9-s + (0.502 − 0.870i)10-s + (−0.0118 − 0.0205i)11-s + (−0.647 − 0.235i)12-s + (−1.12 − 0.945i)13-s + (−0.551 + 0.954i)14-s + (−0.340 − 1.93i)15-s + (0.0434 + 0.246i)16-s + (0.671 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0836 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0836 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.246317 - 0.267861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246317 - 0.267861i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 + 0.684i)T \) |
| 37 | \( 1 + (167. - 150. i)T \) |
| good | 3 | \( 1 + (6.73 - 2.45i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (2.75 - 15.6i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-5.01 + 28.4i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (0.433 + 0.750i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (52.8 + 44.3i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-47.0 + 39.4i)T + (853. - 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-98.8 + 35.9i)T + (5.25e3 - 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-30.1 + 52.2i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-15.3 - 26.5i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 322.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (342. + 287. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 - 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-60.9 + 105. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (42.6 + 241. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-25.1 - 142. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (95.3 + 80.0i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-130. + 740. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-206. + 75.1i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + 377.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-221. + 1.25e3i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (767. - 644. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-168. - 954. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-544. + 942. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.91268296628861611060817709311, −12.18240528444215227191546099479, −11.13709588307510937255209970043, −10.55312776252910291892648760599, −9.911232698354989025084144701445, −7.51026088227201494746512884086, −6.97456047888118191371055348007, −5.17316763931018731478857339208, −3.34631600684073231086137617208, −0.35268028440664967183419199833,
1.52366807069557573994407997996, 5.12432058469207291901141436157, 5.72441022453943488293451867448, 7.36909038223543847743888204931, 8.705762761236021706985548395409, 9.610796100727776390704903338055, 11.42858866323927653495906277561, 12.09960090326785432236527499741, 12.63095437734638902891239259074, 14.59385786294098193589463561402
