| L(s) = 1 | + (0.730 + 1.86i)2-s + (−0.641 − 8.55i)3-s + (−2.93 + 2.72i)4-s + (−16.2 + 11.1i)5-s + (15.4 − 7.44i)6-s + (11.3 + 14.6i)7-s + (−7.20 − 3.47i)8-s + (−46.1 + 6.95i)9-s + (−32.5 − 22.2i)10-s + (28.1 + 4.24i)11-s + (25.1 + 23.3i)12-s + (−51.6 + 64.8i)13-s + (−18.8 + 31.8i)14-s + (105. + 132. i)15-s + (1.19 − 15.9i)16-s + (−76.1 + 23.4i)17-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.658i)2-s + (−0.123 − 1.64i)3-s + (−0.366 + 0.340i)4-s + (−1.45 + 0.993i)5-s + (1.05 − 0.506i)6-s + (0.614 + 0.788i)7-s + (−0.318 − 0.153i)8-s + (−1.70 + 0.257i)9-s + (−1.03 − 0.702i)10-s + (0.772 + 0.116i)11-s + (0.605 + 0.561i)12-s + (−1.10 + 1.38i)13-s + (−0.360 + 0.608i)14-s + (1.81 + 2.27i)15-s + (0.0186 − 0.249i)16-s + (−1.08 + 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.226973 + 0.576931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.226973 + 0.576931i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.730 - 1.86i)T \) |
| 7 | \( 1 + (-11.3 - 14.6i)T \) |
| good | 3 | \( 1 + (0.641 + 8.55i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (16.2 - 11.1i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-28.1 - 4.24i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (51.6 - 64.8i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (76.1 - 23.4i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-11.6 + 20.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (78.6 + 24.2i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (17.9 - 78.5i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (89.9 + 155. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-163. - 151. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (144. + 69.6i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-246. + 118. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-75.3 - 191. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (246. - 228. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (250. + 170. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-430. - 399. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-286. - 496. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (89.5 + 392. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (69.0 - 175. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (60.5 - 104. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-100. - 126. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (671. - 101. i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−14.11285061829488989343360650819, −12.62590524904742903861617122206, −11.79047277125423259837583486384, −11.38506904017310819515480785228, −8.909872177566391393593257999012, −7.80124581010480888415762399137, −7.08096999362933193428510976788, −6.31582133878665649634483296730, −4.32574110741624166610900020531, −2.30771866945052445689973623380,
0.32704097677774630503651726041, 3.59111486090056027445444561455, 4.40588259156950801129160196010, 5.14073158058161669675719569229, 7.76017287387166677837699125671, 8.873533626886707387050859088585, 9.958816242020095736663267737210, 11.01468192746473029530116065819, 11.67787559142767606942799576904, 12.75035217142769978043794874667
