L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (2.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 0.999·15-s + (4 + 6.92i)17-s + (−2 + 3.46i)19-s + (−0.500 − 2.59i)21-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s + 0.999·27-s + 5·29-s + (−3.5 − 6.06i)31-s + (0.499 − 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s − 0.258·15-s + (0.970 + 1.68i)17-s + (−0.458 + 0.794i)19-s + (−0.109 − 0.566i)21-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s + 0.192·27-s + 0.928·29-s + (−0.628 − 1.08i)31-s + (0.0870 − 0.150i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.731740930\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.731740930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−9.659518403218010228369853799927, −8.648337938869299425675133949218, −8.022795475321647779525766668366, −7.38342539216342759655630631574, −6.06543630531402533596344589126, −5.68887996800110533036229011674, −4.63989452383242343099588936017, −3.64120534326883049856286020436, −2.05058978641858857343839272505, −1.32647461576426298406969447795,
0.846239260396293073982028686291, 2.45067608633037747285680715120, 3.46389469796025996116386638006, 4.77944480205365351660651605344, 5.05290141194100084625959097186, 6.36985627647395589409330629028, 7.00829318203567106109216455654, 8.064522442701276376092597899293, 8.742246299390820731005239613668, 9.771575565859570022491580385872