L(s) = 1 | + (−0.809 + 0.587i)2-s + (−1 − 3.07i)3-s + (0.309 − 0.951i)4-s + (−3.11 − 2.26i)5-s + (2.61 + 1.90i)6-s + (−1.19 + 3.66i)7-s + (0.309 + 0.951i)8-s + (−6.04 + 4.39i)9-s + 3.85·10-s + (−0.309 − 3.30i)11-s − 3.23·12-s + (3.23 − 2.35i)13-s + (−1.19 − 3.66i)14-s + (−3.85 + 11.8i)15-s + (−0.809 − 0.587i)16-s + (−1.30 − 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.577 − 1.77i)3-s + (0.154 − 0.475i)4-s + (−1.39 − 1.01i)5-s + (1.06 + 0.776i)6-s + (−0.450 + 1.38i)7-s + (0.109 + 0.336i)8-s + (−2.01 + 1.46i)9-s + 1.21·10-s + (−0.0931 − 0.995i)11-s − 0.934·12-s + (0.897 − 0.652i)13-s + (−0.318 − 0.979i)14-s + (−0.995 + 3.06i)15-s + (−0.202 − 0.146i)16-s + (−0.317 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.309 + 3.30i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (1 + 3.07i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (3.11 + 2.26i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (1.19 - 3.66i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.23 + 2.35i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.30 + 0.951i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 - 0.854T + 23T^{2} \) |
| 29 | \( 1 + (1.76 - 5.42i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.23 + 1.62i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1 - 3.07i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.145 - 0.449i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 + (0.263 + 0.812i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (9.09 - 6.60i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.14 + 6.60i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.927 - 0.673i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 + (-2.23 - 1.62i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.38 - 10.4i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.47 + 5.42i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.73 + 1.26i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + (13.7 - 9.95i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.99695671372290293112477525412, −9.028830481409135347769299651085, −8.347840917476763219714302354061, −8.011123253548584954673435879010, −6.83440790618441897955124767877, −5.91922949582519498270890785764, −5.21873118987883576079609004089, −3.05790286250575764576336802142, −1.26529511227840284060139585803, 0,
3.23843910963168283916880220277, 3.97877204515060317220210693074, 4.49701347906267106458992470949, 6.45280128849526491843161783232, 7.24616582790049482804533822521, 8.393504742109723317316828193058, 9.592921257079716724719451461039, 10.21814967995834433649101753391, 10.94319270031492454655469383823