| L(s) = 1 | + (−9.05 − 9.05i)2-s + (−110. − 266. i)3-s − 1.88e3i·4-s + (−9.90e3 + 4.10e3i)5-s + (−1.41e3 + 3.40e3i)6-s + (−5.25e3 − 2.17e3i)7-s + (−3.56e4 + 3.56e4i)8-s + (6.66e4 − 6.66e4i)9-s + (1.26e5 + 5.25e4i)10-s + (1.69e5 − 4.09e5i)11-s + (−5.01e5 + 2.07e5i)12-s + 2.35e6i·13-s + (2.78e4 + 6.72e4i)14-s + (2.18e6 + 2.18e6i)15-s − 3.21e6·16-s + (5.74e6 + 1.13e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.200 − 0.200i)2-s + (−0.261 − 0.632i)3-s − 0.919i·4-s + (−1.41 + 0.587i)5-s + (−0.0741 + 0.178i)6-s + (−0.118 − 0.0489i)7-s + (−0.384 + 0.384i)8-s + (0.376 − 0.376i)9-s + (0.401 + 0.166i)10-s + (0.317 − 0.765i)11-s + (−0.581 + 0.240i)12-s + 1.75i·13-s + (0.0138 + 0.0334i)14-s + (0.742 + 0.742i)15-s − 0.765·16-s + (0.981 + 0.193i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.281395 + 0.214023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.281395 + 0.214023i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-5.74e6 - 1.13e6i)T \) |
| good | 2 | \( 1 + (9.05 + 9.05i)T + 2.04e3iT^{2} \) |
| 3 | \( 1 + (110. + 266. i)T + (-1.25e5 + 1.25e5i)T^{2} \) |
| 5 | \( 1 + (9.90e3 - 4.10e3i)T + (3.45e7 - 3.45e7i)T^{2} \) |
| 7 | \( 1 + (5.25e3 + 2.17e3i)T + (1.39e9 + 1.39e9i)T^{2} \) |
| 11 | \( 1 + (-1.69e5 + 4.09e5i)T + (-2.01e11 - 2.01e11i)T^{2} \) |
| 13 | \( 1 - 2.35e6iT - 1.79e12T^{2} \) |
| 19 | \( 1 + (-1.13e6 - 1.13e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + (2.16e7 - 5.23e7i)T + (-6.73e14 - 6.73e14i)T^{2} \) |
| 29 | \( 1 + (1.29e7 - 5.35e6i)T + (8.62e15 - 8.62e15i)T^{2} \) |
| 31 | \( 1 + (6.00e7 + 1.44e8i)T + (-1.79e16 + 1.79e16i)T^{2} \) |
| 37 | \( 1 + (1.20e8 + 2.91e8i)T + (-1.25e17 + 1.25e17i)T^{2} \) |
| 41 | \( 1 + (-2.41e8 - 9.99e7i)T + (3.89e17 + 3.89e17i)T^{2} \) |
| 43 | \( 1 + (8.72e8 - 8.72e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.94e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + (-2.38e9 - 2.38e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (2.71e9 - 2.71e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (4.39e9 + 1.81e9i)T + (3.07e19 + 3.07e19i)T^{2} \) |
| 67 | \( 1 + 1.60e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-1.77e9 - 4.27e9i)T + (-1.63e20 + 1.63e20i)T^{2} \) |
| 73 | \( 1 + (-1.95e10 + 8.10e9i)T + (2.21e20 - 2.21e20i)T^{2} \) |
| 79 | \( 1 + (8.31e9 - 2.00e10i)T + (-5.28e20 - 5.28e20i)T^{2} \) |
| 83 | \( 1 + (-8.40e8 - 8.40e8i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 - 8.40e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + (4.16e10 - 1.72e10i)T + (5.05e21 - 5.05e21i)T^{2} \) |
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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
−16.35779798958459942211079643373, −15.12274505802363942874258620722, −13.95167533997136328045513006583, −11.91059808639706883746276061439, −11.26826641113686889644845226374, −9.473333518554913619020374455628, −7.51662193168601250662285195537, −6.21355991209943311677845784032, −3.83563430939999308022160024568, −1.36810306392131784153858490387,
0.18195659706442718272834878937, 3.48753154042823675554333357271, 4.77204452241896054034341552064, 7.44000180637650189339326059218, 8.412394936078243656026336337186, 10.28204197374614291333844501089, 12.00344467648752143024277631804, 12.76229109709372472952791295022, 15.18932921882902453978743309908, 16.04082319280393834890655632364
