L(s) = 1 | + (−0.258 − 0.965i)3-s + (−0.866 − 0.5i)5-s + (−1.67 + 0.965i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)15-s + (1.36 + 1.36i)21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (0.707 + 0.707i)27-s + (1.5 − 0.866i)29-s + 1.93·35-s + (0.866 + 0.5i)41-s + (−1.22 + 0.707i)43-s + 45-s + (0.965 + 1.67i)47-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)3-s + (−0.866 − 0.5i)5-s + (−1.67 + 0.965i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)15-s + (1.36 + 1.36i)21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (0.707 + 0.707i)27-s + (1.5 − 0.866i)29-s + 1.93·35-s + (0.866 + 0.5i)41-s + (−1.22 + 0.707i)43-s + 45-s + (0.965 + 1.67i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5763469573\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5763469573\) |
\(L(1)\) |
|
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 | \( 1 + (0.866 + 0.5i)T \) |
good | 7 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 1.73iT - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.843788557537619042575549634605, −8.234974270061389552289008890163, −7.45378524737999736141143796234, −6.57344752193070641415347748473, −6.16107521862195079120553493082, −5.28106684200324871806824364476, −4.26442323547088705739530216164, −3.08755600352415759791251078898, −2.53699378907030707763002608901, −0.916388439052939729520654807138,
0.50085916346067055312332887919, 2.82135706234538652609662979372, 3.49027925206539228202787407458, 3.97579418128990729079441324801, 4.89905673771802716747096694224, 5.95404109666855116099131355725, 6.79795375833645062813270962591, 7.14661526987059497533873017843, 8.297055335128528254728210896088, 9.038025987448975181428652960928