| L(s) = 1 | + (3.23 − 2.35i)2-s + (2.86 + 2.07i)3-s + (4.94 − 15.2i)4-s − 40.9·5-s + 14.1·6-s + (27.2 − 83.7i)7-s + (−19.7 − 60.8i)8-s + (−71.2 − 219. i)9-s + (−132. + 96.3i)10-s + (115. − 356. i)11-s + (45.7 − 33.2i)12-s + (−289. − 210. i)13-s + (−108. − 335. i)14-s + (−117. − 85.1i)15-s + (−207. − 150. i)16-s + (195. + 601. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.183 + 0.133i)3-s + (0.154 − 0.475i)4-s − 0.732·5-s + 0.160·6-s + (0.209 − 0.646i)7-s + (−0.109 − 0.336i)8-s + (−0.293 − 0.902i)9-s + (−0.419 + 0.304i)10-s + (0.288 − 0.889i)11-s + (0.0917 − 0.0666i)12-s + (−0.475 − 0.345i)13-s + (−0.148 − 0.456i)14-s + (−0.134 − 0.0977i)15-s + (−0.202 − 0.146i)16-s + (0.163 + 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.946138 - 1.59528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946138 - 1.59528i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 31 | \( 1 + (-3.61e3 - 3.94e3i)T \) |
| good | 3 | \( 1 + (-2.86 - 2.07i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + 40.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + (-27.2 + 83.7i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-115. + 356. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (289. + 210. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-195. - 601. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-726. + 528. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (377. + 1.16e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.55e3 - 1.85e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-84.0 + 61.0i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + (-7.39e3 + 5.37e3i)T + (4.54e7 - 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-1.16e4 - 8.46e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-3.03e3 - 9.35e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.62e4 - 1.18e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + 4.65e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (9.91e3 + 3.05e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.64e4 + 8.14e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (9.71e3 + 2.99e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (9.07e3 - 6.59e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.30e4 + 7.10e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.68e4 + 1.13e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.72778650969149046066770566360, −12.39113438621192292330019263015, −11.51397989072550193511109224923, −10.43851515262690564107285928947, −8.996432093365220079680538422373, −7.57528615617718943875307675822, −6.02657429576007538436763880692, −4.27417551595301205710969035437, −3.17833729829382034313911799083, −0.72120779697582523907428531069,
2.37281926384968569649807853527, 4.23913739766441086883788547677, 5.53762414768975388020190267115, 7.27979307713856453512857784241, 8.098348789295154701505390264961, 9.611857798791304775703795442346, 11.43392454359523906536360747883, 12.10458379033934960688604463968, 13.38196618004606878833827892783, 14.47764580682314216583787926954
