L(s) = 1 | + (−0.258 − 0.965i)3-s + (0.707 + 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.866 + 0.499i)9-s + (0.965 − 1.67i)11-s + (0.500 − 0.866i)15-s + (0.258 − 0.965i)21-s + 1.00i·25-s + (0.707 + 0.707i)27-s − 0.517i·29-s + (−0.866 − 0.5i)31-s + (−1.86 − 0.500i)33-s + (0.258 + 0.965i)35-s + (−0.965 − 0.258i)45-s + (0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)3-s + (0.707 + 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.866 + 0.499i)9-s + (0.965 − 1.67i)11-s + (0.500 − 0.866i)15-s + (0.258 − 0.965i)21-s + 1.00i·25-s + (0.707 + 0.707i)27-s − 0.517i·29-s + (−0.866 − 0.5i)31-s + (−1.86 − 0.500i)33-s + (0.258 + 0.965i)35-s + (−0.965 − 0.258i)45-s + (0.499 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.500838556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500838556\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 11 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + 0.517iT - T^{2} \) |
| 31 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
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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
−8.683381503857802813735270014317, −7.960907870930169280011420113447, −7.14861542261471654897831832182, −6.39330175045285116068744716055, −5.79271402896592155731760882381, −5.33992066450791446364137400212, −3.92579579785482447855385132256, −2.90329777599707601540219528673, −2.08048238113798177668263172539, −1.12441256616508916857645788890,
1.31182727849646702546286347396, 2.20798261251585700494844365503, 3.69326227544887647751906656213, 4.43543801068721676699739274423, 4.91377040620799282759007526765, 5.62206043246939965985678359704, 6.61480543349235425251141073352, 7.34483383351005157165239123177, 8.343028830828947732508193184180, 9.120843222359047870505346887782