L(s) = 1 | − 4i·2-s + 11.0i·3-s − 16·4-s + (−42.4 − 36.3i)5-s + 44.3·6-s + 240. i·7-s + 64i·8-s + 120.·9-s + (−145. + 169. i)10-s + 467.·11-s − 177. i·12-s − 75.0i·13-s + 963.·14-s + (402. − 470. i)15-s + 256·16-s − 1.16e3i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.711i·3-s − 0.5·4-s + (−0.759 − 0.649i)5-s + 0.502·6-s + 1.85i·7-s + 0.353i·8-s + 0.494·9-s + (−0.459 + 0.537i)10-s + 1.16·11-s − 0.355i·12-s − 0.123i·13-s + 1.31·14-s + (0.462 − 0.540i)15-s + 0.250·16-s − 0.980i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.6483875903\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6483875903\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 + (42.4 + 36.3i)T \) |
| 23 | \( 1 + 529iT \) |
good | 3 | \( 1 - 11.0iT - 243T^{2} \) |
| 7 | \( 1 - 240. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 467.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 75.0iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.16e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.64e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.19e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.14e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.46e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.67e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.14e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.40e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.74e5iT - 8.58e9T^{2} \) |
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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
−11.91416377598859792138560844014, −10.88498059735110169016034723685, −9.587674224085379069885669026934, −8.991140263873143274583741743480, −8.275694002189865221861418221683, −6.49814334643615339592922618824, −5.06259697136986259754199939664, −4.33701442195461046019889842851, −3.07103450822149430261587834967, −1.59620651067887315376577729516,
0.19880321200947849975181186454, 1.49330827272096297987910942691, 3.85204692396573084825549755858, 4.24051113671857053344454470229, 6.47089814120766112627736680068, 6.87631101648740375325682310331, 7.64400643006395149172263784514, 8.613408336289276844625123099468, 10.20736511868911157295860474724, 10.78045909485283567964469406925