L(s) = 1 | + (1.13 − 5.06i)3-s + 14.6i·5-s − 7i·7-s + (−24.4 − 11.5i)9-s + 13.9·11-s − 45.4·13-s + (74.2 + 16.6i)15-s + 120. i·17-s + 34.9i·19-s + (−35.4 − 7.97i)21-s − 151.·23-s − 89.4·25-s + (−86.3 + 110. i)27-s − 224. i·29-s + 44.2i·31-s + ⋯ |
L(s) = 1 | + (0.219 − 0.975i)3-s + 1.30i·5-s − 0.377i·7-s + (−0.903 − 0.427i)9-s + 0.381·11-s − 0.970·13-s + (1.27 + 0.287i)15-s + 1.72i·17-s + 0.421i·19-s + (−0.368 − 0.0829i)21-s − 1.37·23-s − 0.715·25-s + (−0.615 + 0.787i)27-s − 1.43i·29-s + 0.256i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8869202333\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8869202333\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.13 + 5.06i)T \) |
| 7 | \( 1 + 7iT \) |
good | 5 | \( 1 - 14.6iT - 125T^{2} \) |
| 11 | \( 1 - 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 120. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 34.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 151.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 224. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 44.2iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 459. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 497. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 282. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 48.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 343.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 678. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 820.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 370.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 986. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 484.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 980. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 488.T + 9.12e5T^{2} \) |
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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
−11.48539474731392090602834125894, −10.48899047405576493638244314056, −9.713539451262084619616054099223, −8.210566918188003581358698169622, −7.60791028328402807610479351771, −6.55783802090619345720864945283, −6.01672363458374175634160235737, −4.06210451778192390399491305875, −2.87024661512549964893876811850, −1.70628714633869976902055474491,
0.28976706962254746348352312432, 2.28387101692231050070707661822, 3.78351186823110116960914179723, 4.99293415423498818174125278261, 5.34988635768393002918865294318, 7.09813275014867596252290920237, 8.361243885959425187843071957632, 9.138517056264821923907492002716, 9.619481672226889297282124662238, 10.74989643595067485151895509112