| L(s) = 1 | + (10.3 − 3.36i)2-s + (−9.28 − 12.7i)3-s + (70.0 − 50.9i)4-s + (−25.7 + 49.6i)5-s + (−139. − 101. i)6-s + 71.6i·7-s + (349. − 481. i)8-s + (−2.05 + 6.32i)9-s + (−99.2 + 600. i)10-s + (157. + 485. i)11-s + (−1.30e3 − 423. i)12-s + (218. + 70.9i)13-s + (241. + 742. i)14-s + (873. − 132. i)15-s + (1.14e3 − 3.52e3i)16-s + (−574. + 790. i)17-s + ⋯ |
| L(s) = 1 | + (1.83 − 0.595i)2-s + (−0.595 − 0.820i)3-s + (2.19 − 1.59i)4-s + (−0.459 + 0.887i)5-s + (−1.57 − 1.14i)6-s + 0.552i·7-s + (1.93 − 2.65i)8-s + (−0.00845 + 0.0260i)9-s + (−0.313 + 1.89i)10-s + (0.393 + 1.21i)11-s + (−2.60 − 0.848i)12-s + (0.358 + 0.116i)13-s + (0.328 + 1.01i)14-s + (1.00 − 0.151i)15-s + (1.11 − 3.44i)16-s + (−0.481 + 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.56475 - 1.76781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56475 - 1.76781i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (25.7 - 49.6i)T \) |
| good | 2 | \( 1 + (-10.3 + 3.36i)T + (25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (9.28 + 12.7i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 71.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-157. - 485. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-218. - 70.9i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (574. - 790. i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.47e3 + 1.06e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (2.30e3 - 749. i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.64e3 + 3.37e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.96e3 - 1.42e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.97e3 + 642. i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.35e3 + 4.15e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 8.15e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (8.13e3 + 1.11e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-8.63e3 - 1.18e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (7.24e3 - 2.23e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-3.69e3 - 1.13e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (1.57e4 - 2.16e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-2.58e4 + 1.87e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.40e4 - 4.55e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.19e3 - 2.31e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.08e4 + 8.37e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.83e4 - 8.72e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.96e4 - 2.70e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.52216282449837085726338454195, −14.92106263938693711908502631658, −13.53402392852635197618285221335, −12.29388893836523516627189150706, −11.77321412333421327103072975740, −10.47058656650037166716832577305, −6.97029189082536277725772622366, −6.13131424117909459768433590129, −4.14762285787871067462723451009, −2.14307357779812798699496562340,
3.84408233217511533950040880070, 4.84927887205457905444327991880, 6.21051296827404135194756572547, 8.141789487902534390189004153187, 10.85721927958794837475573783318, 11.86726376074603228140831555762, 13.17248838040287759552145760963, 14.20537225124845184007665652539, 15.73636929164582393095084883661, 16.29330210900036887629145847038
