| L(s) = 1 | + (−1.04 + 3.21i)2-s + (−135. + 98.2i)3-s + (404. + 294. i)4-s + (−656. − 2.01e3i)5-s + (−174. − 537. i)6-s + (−6.76e3 − 4.91e3i)7-s + (−2.77e3 + 2.01e3i)8-s + (2.54e3 − 7.82e3i)9-s + 7.17e3·10-s + (−1.15e4 − 4.71e4i)11-s − 8.36e4·12-s + (2.98e3 − 9.19e3i)13-s + (2.28e4 − 1.66e4i)14-s + (2.86e5 + 2.08e5i)15-s + (7.56e4 + 2.32e5i)16-s + (−1.18e4 − 3.65e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0461 + 0.142i)2-s + (−0.963 + 0.699i)3-s + (0.790 + 0.574i)4-s + (−0.469 − 1.44i)5-s + (−0.0549 − 0.169i)6-s + (−1.06 − 0.774i)7-s + (−0.239 + 0.173i)8-s + (0.129 − 0.397i)9-s + 0.227·10-s + (−0.238 − 0.971i)11-s − 1.16·12-s + (0.0290 − 0.0893i)13-s + (0.159 − 0.115i)14-s + (1.46 + 1.06i)15-s + (0.288 + 0.887i)16-s + (−0.0345 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.143254 - 0.290930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143254 - 0.290930i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.15e4 + 4.71e4i)T \) |
| good | 2 | \( 1 + (1.04 - 3.21i)T + (-414. - 300. i)T^{2} \) |
| 3 | \( 1 + (135. - 98.2i)T + (6.08e3 - 1.87e4i)T^{2} \) |
| 5 | \( 1 + (656. + 2.01e3i)T + (-1.58e6 + 1.14e6i)T^{2} \) |
| 7 | \( 1 + (6.76e3 + 4.91e3i)T + (1.24e7 + 3.83e7i)T^{2} \) |
| 13 | \( 1 + (-2.98e3 + 9.19e3i)T + (-8.57e9 - 6.23e9i)T^{2} \) |
| 17 | \( 1 + (1.18e4 + 3.65e4i)T + (-9.59e10 + 6.97e10i)T^{2} \) |
| 19 | \( 1 + (7.78e5 - 5.65e5i)T + (9.97e10 - 3.06e11i)T^{2} \) |
| 23 | \( 1 + 1.41e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (-2.76e6 - 2.00e6i)T + (4.48e12 + 1.37e13i)T^{2} \) |
| 31 | \( 1 + (-1.83e6 + 5.64e6i)T + (-2.13e13 - 1.55e13i)T^{2} \) |
| 37 | \( 1 + (-4.20e6 - 3.05e6i)T + (4.01e13 + 1.23e14i)T^{2} \) |
| 41 | \( 1 + (-1.36e7 + 9.94e6i)T + (1.01e14 - 3.11e14i)T^{2} \) |
| 43 | \( 1 + 2.68e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-8.77e6 + 6.37e6i)T + (3.45e14 - 1.06e15i)T^{2} \) |
| 53 | \( 1 + (-3.03e6 + 9.34e6i)T + (-2.66e15 - 1.93e15i)T^{2} \) |
| 59 | \( 1 + (-9.47e7 - 6.88e7i)T + (2.67e15 + 8.23e15i)T^{2} \) |
| 61 | \( 1 + (1.67e7 + 5.15e7i)T + (-9.46e15 + 6.87e15i)T^{2} \) |
| 67 | \( 1 - 7.74e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + (2.65e7 + 8.17e7i)T + (-3.70e16 + 2.69e16i)T^{2} \) |
| 73 | \( 1 + (2.92e8 + 2.12e8i)T + (1.81e16 + 5.59e16i)T^{2} \) |
| 79 | \( 1 + (1.24e8 - 3.82e8i)T + (-9.69e16 - 7.04e16i)T^{2} \) |
| 83 | \( 1 + (2.38e8 + 7.32e8i)T + (-1.51e17 + 1.09e17i)T^{2} \) |
| 89 | \( 1 - 4.70e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-8.00e7 + 2.46e8i)T + (-6.15e17 - 4.46e17i)T^{2} \) |
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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
−17.00087915262096303527712273280, −16.47192246719794633584534486765, −15.87002266838155727323185919990, −13.04393821261702774414574320185, −11.83986244488961013924763487183, −10.37804811293276978865746476477, −8.250278432551013337685368641385, −6.04983064497897663735247434978, −4.03287776664877316402295816778, −0.20146895333396956150182450759,
2.52961569554633417634546036154, 6.24554042178348428316128668814, 6.92007537535108538680794844682, 10.17604560310738336882935929667, 11.43088247854714393760388716017, 12.52141799117755306813590989648, 14.88223317319871584140016305135, 15.82907539881016334666998729298, 17.80517530021550228453153223616, 18.84634648909945663217686680518
