| 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
−14.59385786294098193589463561402, −12.63095437734638902891239259074, −12.09960090326785432236527499741, −11.42858866323927653495906277561, −9.610796100727776390704903338055, −8.705762761236021706985548395409, −7.36909038223543847743888204931, −5.72441022453943488293451867448, −5.12432058469207291901141436157, −1.52366807069557573994407997996,
0.35268028440664967183419199833, 3.34631600684073231086137617208, 5.17316763931018731478857339208, 6.97456047888118191371055348007, 7.51026088227201494746512884086, 9.911232698354989025084144701445, 10.55312776252910291892648760599, 11.13709588307510937255209970043, 12.18240528444215227191546099479, 13.91268296628861611060817709311
