L(s) = 1 | + (1.22 − 1.22i)3-s + 2.44·5-s + (1 − 2.44i)7-s − 2.99i·9-s + 2.44i·13-s + (2.99 − 2.99i)15-s + 4.89·17-s + 2.44i·19-s + (−1.77 − 4.22i)21-s − 6i·23-s + 0.999·25-s + (−3.67 − 3.67i)27-s + 6i·29-s + (2.44 − 5.99i)35-s + 2·37-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)3-s + 1.09·5-s + (0.377 − 0.925i)7-s − 0.999i·9-s + 0.679i·13-s + (0.774 − 0.774i)15-s + 1.18·17-s + 0.561i·19-s + (−0.387 − 0.921i)21-s − 1.25i·23-s + 0.199·25-s + (−0.707 − 0.707i)27-s + 1.11i·29-s + (0.414 − 1.01i)35-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.741847515\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.741847515\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−9.512082642079016747906110700733, −8.573994528057141721048411346874, −7.86595516989425928118252501410, −6.99570394146391909433856776171, −6.36062411835278405477238570711, −5.37598881321753188306594533393, −4.20186088531844883930035889134, −3.16954350753048597522444517988, −1.99038919907473091136245303705, −1.16417571717573110167605445649,
1.69131032866680340528989610033, 2.63687544521747585790303227821, 3.50211278522476471249498454669, 4.87588054465440716632789979036, 5.48528843572093334465524675341, 6.19015941168699676005615599687, 7.68877592687873002278933325824, 8.146533330175561852106786406887, 9.324232321219682946150939364573, 9.489220541971416473657323253942