| L(s) = 1 | + (−1.68 + 1.08i)2-s + (−0.426 + 2.96i)3-s + (1.66 − 3.63i)4-s + (−11.8 + 3.47i)5-s + (−2.49 − 5.45i)6-s + (13.0 + 15.0i)7-s + (1.13 + 7.91i)8-s + (−8.63 − 2.53i)9-s + (16.1 − 18.6i)10-s + (−32.9 − 21.1i)11-s + (10.0 + 6.48i)12-s + (−10.0 + 11.5i)13-s + (−38.2 − 11.2i)14-s + (−5.26 − 36.6i)15-s + (−10.4 − 12.0i)16-s + (−25.9 − 56.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−1.05 + 0.310i)5-s + (−0.169 − 0.371i)6-s + (0.704 + 0.813i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.510 − 0.589i)10-s + (−0.903 − 0.580i)11-s + (0.242 + 0.156i)12-s + (−0.213 + 0.246i)13-s + (−0.729 − 0.214i)14-s + (−0.0905 − 0.630i)15-s + (−0.163 − 0.188i)16-s + (−0.370 − 0.811i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0203350 - 0.0299061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0203350 - 0.0299061i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.68 - 1.08i)T \) |
| 3 | \( 1 + (0.426 - 2.96i)T \) |
| 23 | \( 1 + (-5.55 + 110. i)T \) |
| good | 5 | \( 1 + (11.8 - 3.47i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (-13.0 - 15.0i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (32.9 + 21.1i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (10.0 - 11.5i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (25.9 + 56.8i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-14.2 + 31.2i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (26.7 + 58.5i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (27.1 + 188. i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (144. + 42.5i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (378. - 111. i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (19.5 - 135. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 - 87.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-419. - 483. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (578. - 667. i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-44.5 - 310. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (24.7 - 15.8i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (638. - 410. i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-275. + 603. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (439. - 506. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (556. + 163. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-130. + 908. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (643. - 188. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
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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
−11.92375270665572613148246678991, −11.33507986771545158429323005663, −10.39527537686412614447999223691, −9.048267895220421833294621049283, −8.231647377143772945420160954020, −7.25179668683868083300990474498, −5.68491845060384311683852258782, −4.52370720361835633690096606009, −2.70546751380966425540270189213, −0.02188123879615427756100435250,
1.64507626931804508150166432835, 3.63073572733997512697625524606, 5.04533670213027720841344686536, 7.06880677623245472718988417281, 7.80428410259975870472460164637, 8.529547580488677292121024963103, 10.17197600631024985678184545434, 11.00188721550857144513381011913, 11.97531145254668969794488207717, 12.74159554042854436191461088345
