| L(s) = 1 | + (0.694 − 3.93i)2-s + (0.515 + 2.92i)3-s + (−15.0 − 5.47i)4-s + (33.8 − 28.4i)5-s + 11.8·6-s + (−11.2 + 9.41i)7-s + (−32 + 55.4i)8-s + (220. − 80.0i)9-s + (−88.4 − 153. i)10-s + (126. − 219. i)11-s + (8.25 − 46.8i)12-s + (−946. − 344. i)13-s + (29.2 + 50.7i)14-s + (100. + 84.5i)15-s + (196. + 164. i)16-s + (964. − 350. i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.0331 + 0.187i)3-s + (−0.469 − 0.171i)4-s + (0.606 − 0.508i)5-s + 0.134·6-s + (−0.0865 + 0.0726i)7-s + (−0.176 + 0.306i)8-s + (0.905 − 0.329i)9-s + (−0.279 − 0.484i)10-s + (0.315 − 0.546i)11-s + (0.0165 − 0.0938i)12-s + (−1.55 − 0.565i)13-s + (0.0399 + 0.0691i)14-s + (0.115 + 0.0969i)15-s + (0.191 + 0.160i)16-s + (0.809 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.888291 - 1.55401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.888291 - 1.55401i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.694 + 3.93i)T \) |
| 37 | \( 1 + (-7.09e3 + 4.35e3i)T \) |
| good | 3 | \( 1 + (-0.515 - 2.92i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (-33.8 + 28.4i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (11.2 - 9.41i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (-126. + 219. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (946. + 344. i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-964. + 350. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 19 | \( 1 + (439. + 2.49e3i)T + (-2.32e6 + 8.46e5i)T^{2} \) |
| 23 | \( 1 + (1.21e3 + 2.11e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-632. + 1.09e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 7.99e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-8.28e3 - 3.01e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 - 1.93e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.37e3 + 4.11e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.77e4 - 2.32e4i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-2.77e4 - 2.33e4i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (3.45e4 + 1.25e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-2.82e4 + 2.36e4i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-664. - 3.76e3i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 - 2.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (6.37e4 - 5.35e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (7.53e4 - 2.74e4i)T + (3.01e9 - 2.53e9i)T^{2} \) |
| 89 | \( 1 + (975. + 818. i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-1.76e4 - 3.04e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−12.92882761633787538028340305195, −12.36945110083821300026547830483, −10.93838081283913803233195568722, −9.752954456501806916185890601935, −9.125058324796629565073767267877, −7.35534231573285204372814518016, −5.59307053595745555001323378616, −4.35414549245665678366444882685, −2.57527380728718102148733515558, −0.77617896411557860769026919182,
1.94425887290083469188980147716, 4.10611194238689670874871980436, 5.64411111787295122048977594226, 6.96536883800158488561411339527, 7.77275905424391628275048900786, 9.618971725858408303088314244469, 10.19276667089328638196182205959, 12.09778386156997970445526404723, 12.94099080368246573758449888039, 14.34554812279019022343441239192
