| L(s) = 1 | + (−15.0 − 5.47i)2-s + (−25.7 − 146. i)3-s + (196. + 164. i)4-s + (−400. + 335. i)5-s + (−412. + 2.33e3i)6-s + (3.63e3 − 6.30e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (−2.20e3 + 801. i)9-s + (7.85e3 − 2.85e3i)10-s + (−1.83e4 − 3.17e4i)11-s + (1.89e4 − 3.29e4i)12-s + (2.96e4 − 1.67e5i)13-s + (−8.91e4 + 7.48e4i)14-s + (5.94e4 + 4.98e4i)15-s + (1.13e4 + 6.45e4i)16-s + (2.87e5 + 1.04e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.183 − 1.04i)3-s + (0.383 + 0.321i)4-s + (−0.286 + 0.240i)5-s + (−0.129 + 0.736i)6-s + (0.572 − 0.992i)7-s + (−0.176 − 0.306i)8-s + (−0.111 + 0.0407i)9-s + (0.248 − 0.0904i)10-s + (−0.377 − 0.654i)11-s + (0.264 − 0.458i)12-s + (0.287 − 1.63i)13-s + (−0.620 + 0.520i)14-s + (0.303 + 0.254i)15-s + (0.0434 + 0.246i)16-s + (0.833 + 0.303i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.112236 + 0.827937i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112236 + 0.827937i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.0 + 5.47i)T \) |
| 19 | \( 1 + (-6.71e4 - 5.64e5i)T \) |
| good | 3 | \( 1 + (25.7 + 146. i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (400. - 335. i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (-3.63e3 + 6.30e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (1.83e4 + 3.17e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-2.96e4 + 1.67e5i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-2.87e5 - 1.04e5i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (1.45e6 + 1.22e6i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (6.50e6 - 2.36e6i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-1.23e6 + 2.14e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 4.51e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-5.47e6 - 3.10e7i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-7.79e6 + 6.54e6i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-2.07e7 + 7.54e6i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (6.86e7 + 5.75e7i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (-1.01e6 - 3.69e5i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (2.97e7 + 2.49e7i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (7.05e7 - 2.56e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (-2.46e7 + 2.06e7i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (5.73e7 + 3.25e8i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (4.63e7 + 2.62e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (-7.08e7 + 1.22e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (1.35e7 - 7.69e7i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (3.41e7 + 1.24e7i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−13.37580797533368496754475250631, −12.41693829735872388728488794838, −11.07904070847644283118027868536, −10.16857881612826382754453546920, −7.970259177715206538385620518286, −7.63133445283422355318956638952, −5.93826594352159979395641257940, −3.49836309098174840757520439459, −1.49593407863165118234391954592, −0.40089490855036206186542766520,
1.94979218216208680340024845770, 4.27628683342018813600159841210, 5.55031393057313856940228702768, 7.45367211450248372675056330588, 8.950282183670247598533797127322, 9.749775318880386265703375739523, 11.17579215884396368881207475717, 12.10591622425060156013189637696, 14.13968543943971146765783753779, 15.45614628716010398358579546117
