| L(s) = 1 | + (1.04 − 4.57i)2-s + (3.32 − 4.16i)3-s + (−12.6 − 6.06i)4-s + (−4.03 + 5.06i)5-s + (−15.5 − 19.5i)6-s + (18.4 + 1.16i)7-s + (−17.5 + 21.9i)8-s + (−0.310 − 1.36i)9-s + (18.9 + 23.7i)10-s + (−4.70 + 20.6i)11-s + (−67.1 + 32.3i)12-s + (7.08 − 31.0i)13-s + (24.6 − 83.2i)14-s + (7.67 + 33.6i)15-s + (12.3 + 15.4i)16-s + (37.8 − 18.2i)17-s + ⋯ |
| L(s) = 1 | + (0.368 − 1.61i)2-s + (0.639 − 0.801i)3-s + (−1.57 − 0.758i)4-s + (−0.361 + 0.452i)5-s + (−1.06 − 1.32i)6-s + (0.998 + 0.0628i)7-s + (−0.773 + 0.970i)8-s + (−0.0115 − 0.0504i)9-s + (0.598 + 0.750i)10-s + (−0.128 + 0.565i)11-s + (−1.61 + 0.777i)12-s + (0.151 − 0.662i)13-s + (0.469 − 1.58i)14-s + (0.132 + 0.579i)15-s + (0.192 + 0.241i)16-s + (0.540 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.522205 - 1.75162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.522205 - 1.75162i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-18.4 - 1.16i)T \) |
| good | 2 | \( 1 + (-1.04 + 4.57i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-3.32 + 4.16i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (4.03 - 5.06i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (4.70 - 20.6i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-7.08 + 31.0i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-37.8 + 18.2i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-117. - 56.7i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-251. + 121. i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (270. - 130. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (130. - 163. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (69.2 + 86.8i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (71.8 - 315. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-90.8 - 43.7i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (155. + 194. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (532. - 256. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-122. - 59.1i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (146. + 642. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (296. + 1.29e3i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (136. + 599. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 220.T + 9.12e5T^{2} \) |
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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
−14.24449739693857198457046136417, −13.22159920330696134881755013929, −12.36453637247043436849329959664, −11.21802186656160705567995996524, −10.32913922728512167584420376696, −8.608851597407316057947012331110, −7.36076943298705849011478924824, −4.80125925809523301846221165922, −3.01112714772551625749453103406, −1.62288686729963881626325871616,
4.04024900443574914668189952666, 5.06487441997841913802047207606, 6.78372347308692679719726023305, 8.443656234072542531925675691819, 8.716053338516610490363136073804, 10.70214418326544577816272900397, 12.48852355308750126630466841048, 14.02997967871551224794333412586, 14.61589058999221926127982957867, 15.47542247107780649644819825018
