| L(s) = 1 | + (4.90 + 26.5i)3-s + (−9.93 − 5.73i)5-s + (−128. − 221. i)7-s + (−680. + 260. i)9-s + (−1.22e3 + 707. i)11-s + (475. − 822. i)13-s + (103. − 291. i)15-s − 6.77e3i·17-s − 2.46e3·19-s + (5.25e3 − 4.48e3i)21-s + (1.11e3 + 645. i)23-s + (−7.74e3 − 1.34e4i)25-s + (−1.02e4 − 1.68e4i)27-s + (−1.02e3 + 591. i)29-s + (8.78e3 − 1.52e4i)31-s + ⋯ |
| L(s) = 1 | + (0.181 + 0.983i)3-s + (−0.0794 − 0.0458i)5-s + (−0.373 − 0.646i)7-s + (−0.934 + 0.357i)9-s + (−0.920 + 0.531i)11-s + (0.216 − 0.374i)13-s + (0.0306 − 0.0864i)15-s − 1.37i·17-s − 0.359·19-s + (0.567 − 0.484i)21-s + (0.0918 + 0.0530i)23-s + (−0.495 − 0.858i)25-s + (−0.520 − 0.853i)27-s + (−0.0420 + 0.0242i)29-s + (0.294 − 0.510i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.184896 - 0.314355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184896 - 0.314355i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.90 - 26.5i)T \) |
| good | 5 | \( 1 + (9.93 + 5.73i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (128. + 221. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.22e3 - 707. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-475. + 822. i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 6.77e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.46e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.11e3 - 645. i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.02e3 - 591. i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-8.78e3 + 1.52e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 8.19e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (4.63e4 + 2.67e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.63e3 - 6.30e3i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.20e4 - 6.95e3i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.87e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.88e5 - 1.66e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (5.83e4 + 1.01e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.72e5 - 2.98e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 6.01e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 5.63e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (2.98e4 + 5.16e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (4.19e5 - 2.42e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 4.33e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (8.33e5 + 1.44e6i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−13.25150542267815911693874699465, −11.80758469114970834701859956028, −10.51418247218100753918124285958, −9.874724468825844587410250252109, −8.534708614475081304349900151494, −7.21677373307687036880046043239, −5.43087814098459649847226613208, −4.20154600231287236723184772745, −2.75169817213710436014870428935, −0.12921012432249614192555710708,
1.82485738460621496808006692307, 3.30659522336859212786373070252, 5.55632355644919709686447637907, 6.63371911922036387597864013278, 8.042127551453430396268444867737, 8.900666478904792718336349164858, 10.54677869195768720551378244079, 11.78191486814226190450055104927, 12.79544463940855875533949995280, 13.53561726139647104316298528048
