| L(s) = 1 | + (−15.0 + 5.47i)2-s + (34.1 − 193. i)3-s + (196. − 164. i)4-s + (1.83e3 + 1.54e3i)5-s + (547. + 3.10e3i)6-s + (977. + 1.69e3i)7-s + (−2.04e3 + 3.54e3i)8-s + (−1.79e4 − 6.53e3i)9-s + (−3.60e4 − 1.31e4i)10-s + (−4.01e4 + 6.95e4i)11-s + (−2.52e4 − 4.36e4i)12-s + (3.82e3 + 2.16e4i)13-s + (−2.39e4 − 2.01e4i)14-s + (3.62e5 − 3.03e5i)15-s + (1.13e4 − 6.45e4i)16-s + (−3.32e5 + 1.20e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.243 − 1.38i)3-s + (0.383 − 0.321i)4-s + (1.31 + 1.10i)5-s + (0.172 + 0.977i)6-s + (0.153 + 0.266i)7-s + (−0.176 + 0.306i)8-s + (−0.912 − 0.331i)9-s + (−1.14 − 0.415i)10-s + (−0.827 + 1.43i)11-s + (−0.350 − 0.607i)12-s + (0.0371 + 0.210i)13-s + (−0.166 − 0.139i)14-s + (1.84 − 1.54i)15-s + (0.0434 − 0.246i)16-s + (−0.965 + 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.33777 + 0.767250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33777 + 0.767250i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.0 - 5.47i)T \) |
| 19 | \( 1 + (-3.45e5 - 4.51e5i)T \) |
| good | 3 | \( 1 + (-34.1 + 193. i)T + (-1.84e4 - 6.73e3i)T^{2} \) |
| 5 | \( 1 + (-1.83e3 - 1.54e3i)T + (3.39e5 + 1.92e6i)T^{2} \) |
| 7 | \( 1 + (-977. - 1.69e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.01e4 - 6.95e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-3.82e3 - 2.16e4i)T + (-9.96e9 + 3.62e9i)T^{2} \) |
| 17 | \( 1 + (3.32e5 - 1.20e5i)T + (9.08e10 - 7.62e10i)T^{2} \) |
| 23 | \( 1 + (1.58e6 - 1.33e6i)T + (3.12e11 - 1.77e12i)T^{2} \) |
| 29 | \( 1 + (5.68e5 + 2.06e5i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (-1.78e6 - 3.08e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 5.87e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-5.06e6 + 2.87e7i)T + (-3.07e14 - 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-1.75e7 - 1.46e7i)T + (8.72e13 + 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-1.46e7 - 5.31e6i)T + (8.57e14 + 7.19e14i)T^{2} \) |
| 53 | \( 1 + (1.17e7 - 9.89e6i)T + (5.72e14 - 3.24e15i)T^{2} \) |
| 59 | \( 1 + (7.36e7 - 2.68e7i)T + (6.63e15 - 5.56e15i)T^{2} \) |
| 61 | \( 1 + (-3.63e7 + 3.05e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (-2.22e8 - 8.10e7i)T + (2.08e16 + 1.74e16i)T^{2} \) |
| 71 | \( 1 + (2.67e8 + 2.24e8i)T + (7.96e15 + 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-8.05e7 + 4.56e8i)T + (-5.53e16 - 2.01e16i)T^{2} \) |
| 79 | \( 1 + (7.32e7 - 4.15e8i)T + (-1.12e17 - 4.09e16i)T^{2} \) |
| 83 | \( 1 + (-2.22e8 - 3.84e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-1.68e7 - 9.56e7i)T + (-3.29e17 + 1.19e17i)T^{2} \) |
| 97 | \( 1 + (5.56e8 - 2.02e8i)T + (5.82e17 - 4.88e17i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31875433124965574129164538637, −13.45954149397219834618423213083, −12.21020072712848964204090183825, −10.56217393497830508471364303188, −9.520588107929010187691099056161, −7.78191816252313238003459297191, −6.90262790289871137100757111213, −5.81995810186216284556374288427, −2.33867901750170591056214719028, −1.75911080171809343108991523521,
0.65289955200938061046318272690, 2.62511750792484643143324227666, 4.53825830444426722686400963910, 5.83346305441774937196858859696, 8.372661937058383584900562005203, 9.233071504386708480185578387190, 10.13249893747715954499676846376, 11.13836914414816499202354530943, 13.06076708040815833897034765528, 14.01524749064885120036113340953
