| L(s) = 1 | + (−2.55 + 6.16i)2-s + (−1.90 + 2.84i)3-s + (−20.2 − 20.2i)4-s + (−0.0751 − 0.378i)5-s + (−12.7 − 19.0i)6-s + (−11.8 + 59.4i)7-s + (77.6 − 32.1i)8-s + (26.5 + 64.0i)9-s + (2.52 + 0.502i)10-s + (165. − 110. i)11-s + (95.9 − 19.0i)12-s + (−155. + 155. i)13-s + (−336. − 224. i)14-s + (1.21 + 0.504i)15-s + 103. i·16-s + (288. − 3.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.638 + 1.54i)2-s + (−0.211 + 0.316i)3-s + (−1.26 − 1.26i)4-s + (−0.00300 − 0.0151i)5-s + (−0.352 − 0.528i)6-s + (−0.241 + 1.21i)7-s + (1.21 − 0.502i)8-s + (0.327 + 0.790i)9-s + (0.0252 + 0.00502i)10-s + (1.36 − 0.911i)11-s + (0.666 − 0.132i)12-s + (−0.922 + 0.922i)13-s + (−1.71 − 1.14i)14-s + (0.00541 + 0.00224i)15-s + 0.405i·16-s + (0.999 − 0.0105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0819367 + 0.750018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0819367 + 0.750018i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-288. + 3.04i)T \) |
| good | 2 | \( 1 + (2.55 - 6.16i)T + (-11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (1.90 - 2.84i)T + (-30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (0.0751 + 0.378i)T + (-577. + 239. i)T^{2} \) |
| 7 | \( 1 + (11.8 - 59.4i)T + (-2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (-165. + 110. i)T + (5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (155. - 155. i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (-68.0 + 164. i)T + (-9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-15.8 - 23.6i)T + (-1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (178. - 35.4i)T + (6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (126. + 84.5i)T + (3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (899. - 1.34e3i)T + (-7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (359. - 1.80e3i)T + (-2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (882. + 2.13e3i)T + (-2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-1.19e3 + 1.19e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.07e3 + 2.60e3i)T + (-5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-355. + 147. i)T + (8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-4.94e3 - 983. i)T + (1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 + 5.34e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-2.67e3 + 4.00e3i)T + (-9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-963. - 4.84e3i)T + (-2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (5.04e3 - 3.36e3i)T + (1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (-1.12e4 - 4.65e3i)T + (3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (5.71e3 + 5.71e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (3.65e3 - 727. i)T + (8.17e7 - 3.38e7i)T^{2} \) |
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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
−18.60491922736840012036352846322, −16.90703094605841214280231944753, −16.41804539872205332276169676867, −15.13416503330818274294303119076, −14.06141992137308582889742753851, −11.84775987415940263267070291660, −9.652968706143243445461354797938, −8.557103248163052676826944978372, −6.76593479522242079502128308980, −5.27338845524166203920464575402,
1.03504754520796436563192874866, 3.75411617890158317755406799094, 7.23849662323496730401544187259, 9.463401943653178739612171824824, 10.38775131898577763506064508548, 12.01464566238774629928113938749, 12.76109361777317640708335220197, 14.59120014910894004760784639923, 17.01690521063373148676583024252, 17.70977648146141868748230497450
