| L(s) = 1 | + (−0.847 + 0.489i)2-s + (−63.5 + 110. i)4-s + 246. i·5-s + (−1.13e3 − 655. i)7-s − 249. i·8-s + (−120. − 208. i)10-s + (−6.35e3 + 3.67e3i)11-s + (4.39e3 − 6.59e3i)13-s + 1.28e3·14-s + (−8.00e3 − 1.38e4i)16-s + (6.09e3 − 1.05e4i)17-s + (−2.65e4 − 1.53e4i)19-s + (−2.70e4 − 1.56e4i)20-s + (3.59e3 − 6.22e3i)22-s + (5.23e4 + 9.06e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.0749 + 0.0432i)2-s + (−0.496 + 0.859i)4-s + 0.880i·5-s + (−1.25 − 0.722i)7-s − 0.172i·8-s + (−0.0380 − 0.0659i)10-s + (−1.44 + 0.831i)11-s + (0.554 − 0.831i)13-s + 0.125·14-s + (−0.488 − 0.846i)16-s + (0.301 − 0.521i)17-s + (−0.887 − 0.512i)19-s + (−0.756 − 0.436i)20-s + (0.0719 − 0.124i)22-s + (0.896 + 1.55i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6603351092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6603351092\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (-4.39e3 + 6.59e3i)T \) |
| good | 2 | \( 1 + (0.847 - 0.489i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 - 246. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (1.13e3 + 655. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (6.35e3 - 3.67e3i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-6.09e3 + 1.05e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.65e4 + 1.53e4i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-5.23e4 - 9.06e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.07e4 - 7.06e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.66e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (2.88e5 - 1.66e5i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.02e5 + 5.92e4i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-4.29e5 + 7.44e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 6.89e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.86e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.16e6 + 6.71e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (3.49e5 - 6.05e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.38e6 + 8.00e5i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.31e5 - 7.59e4i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 6.04e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.92e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.18e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (8.33e6 - 4.81e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.09e6 - 6.30e5i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.34733268597672908632665947649, −10.71907421104830978756325238121, −10.12481801240209204065379238267, −8.845951914262637926612388651560, −7.43570045879906066951758196672, −6.93612667531326705797628787637, −5.15828282224080951105346723972, −3.53054979911807512108359406391, −2.83659858208026386274589950489, −0.28375044486559289253789360626,
0.811272428150209520997149672563, 2.54688145827752690800664166479, 4.31496665629942797715529257835, 5.61010065974160136275503032031, 6.32180812163208516006924959944, 8.408091717911459498895481135551, 8.984651180315677043214819987322, 10.08376706773297410023919966917, 11.01481610545910374434931930094, 12.71891666079000339904342176856
