| L(s) = 1 | + (2 − 3.46i)2-s + (−4.5 − 7.79i)3-s + (−7.99 − 13.8i)4-s + (−2.25 + 3.89i)5-s − 36·6-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (9.00 + 15.5i)10-s + (58.0 + 100. i)11-s + (−72 + 124. i)12-s − 85.4·13-s + 40.5·15-s + (−128 + 221. i)16-s + (−16.6 − 28.8i)17-s + (162 + 280. i)18-s + (−317. + 550. i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.0402 + 0.0697i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.0284 + 0.0493i)10-s + (0.144 + 0.250i)11-s + (−0.144 + 0.249i)12-s − 0.140·13-s + 0.0465·15-s + (−0.125 + 0.216i)16-s + (−0.0139 − 0.0241i)17-s + (0.117 + 0.204i)18-s + (−0.201 + 0.349i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.799269675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799269675\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (2.25 - 3.89i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-58.0 - 100. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 85.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + (16.6 + 28.8i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (317. - 550. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.36e3 - 2.36e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 5.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (139. + 241. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.51e3 - 2.63e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 819.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.70e3 + 6.41e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-6.84e3 - 1.18e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.11e4 + 1.93e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-6.34e3 + 1.09e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.61e4 - 4.52e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (3.84e4 + 6.66e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.67e4 + 2.90e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 6.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (4.60e4 - 7.98e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.52e5T + 8.58e9T^{2} \) |
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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
−11.03334347804853952334087444300, −10.14440075211043644268429962337, −9.139543682253887159196510423875, −7.952904241711012204984593614001, −6.87432692669783178625408153198, −5.81885558169677874373478035254, −4.75816470690841193528849070244, −3.48574176551513432328104320130, −2.16989529920840359056549155876, −0.978045514965047791858217735623,
0.55655967611740390127177039663, 2.65557472819995092016172701871, 4.04853548720884862749081578565, 4.88470861381797690758866630384, 6.02922364482775022172083533449, 6.85144687383743232668091520067, 8.147460903875689994298280091092, 8.933487492209771430374042034747, 10.06673701932031822857259955901, 10.94856591008334128075412753255
