L(s) = 1 | − 3-s + 1.73i·5-s − 2·9-s − 1.73i·11-s − 1.73i·15-s + 5.19i·17-s − 7·19-s − 8.66i·23-s + 2.00·25-s + 5·27-s − 6·29-s − 5·31-s + 1.73i·33-s − 5·37-s − 6.92i·41-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.774i·5-s − 0.666·9-s − 0.522i·11-s − 0.447i·15-s + 1.26i·17-s − 1.60·19-s − 1.80i·23-s + 0.400·25-s + 0.962·27-s − 1.11·29-s − 0.898·31-s + 0.301i·33-s − 0.821·37-s − 1.08i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + 8.66iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 8.66iT - 61T^{2} \) |
| 67 | \( 1 - 5.19iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 12.1iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−10.35604132320528957203170963425, −8.811387563063235717931825421809, −8.424139223477214800317221720020, −7.10152964215978916313235657320, −6.29576725602317367963590051517, −5.72246113567273475562701106067, −4.43140932129377275546216121694, −3.32709934635960876556728387188, −2.12425479085155313175740843581, 0,
1.73270708803653011484508016380, 3.24897452515178296615306938291, 4.62396189926426788124148700380, 5.25680477505961232193244877553, 6.18300725804138709988492524008, 7.20485023568168009974022485961, 8.147538782730181230325227529983, 9.112175241764027222849099826751, 9.643056866037542507659769792663