L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.499 + 0.866i)4-s + 1.41i·11-s + (0.499 − 0.866i)16-s + (1.00 − 1.73i)22-s − 1.41i·23-s + 25-s + (−1.22 + 0.707i)29-s + (−1.22 + 0.707i)32-s + (−1.22 + 0.707i)44-s + (−1.00 + 1.73i)46-s + (−1.22 − 0.707i)50-s + (1.22 + 0.707i)53-s + 2·58-s + 0.999·64-s + ⋯ |
L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.499 + 0.866i)4-s + 1.41i·11-s + (0.499 − 0.866i)16-s + (1.00 − 1.73i)22-s − 1.41i·23-s + 25-s + (−1.22 + 0.707i)29-s + (−1.22 + 0.707i)32-s + (−1.22 + 0.707i)44-s + (−1.00 + 1.73i)46-s + (−1.22 − 0.707i)50-s + (1.22 + 0.707i)53-s + 2·58-s + 0.999·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5821774161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5821774161\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−8.762745365432376273881345224011, −8.236134793495151317261377067565, −7.21952114621616392550246030328, −6.94704489609035353981663165135, −5.66186782385849580269429033617, −4.85539147311880201060511145956, −3.99701773176668348038927274079, −2.75948145275648521759044180893, −2.12283783571082670787491667006, −1.09195187999228471846735585429,
0.58779138915295839115903035492, 1.76577461545252241622351190108, 3.18974392243905934361556201746, 3.85379958417569450328375084324, 5.16374134524701658411467613424, 5.91509545421500459905379293666, 6.49947191463816452622312322505, 7.41815542299176498850375685173, 7.84282387628413785786680644402, 8.672957281809207667144861596026