| L(s) = 1 | − 0.644·2-s + 2.95·3-s − 1.58·4-s + 4.36·5-s − 1.90·6-s + 2.31·8-s + 5.73·9-s − 2.81·10-s − 4.65·11-s − 4.68·12-s + 0.769·13-s + 12.9·15-s + 1.67·16-s − 0.371·17-s − 3.69·18-s + 1.98·19-s − 6.91·20-s + 3.00·22-s + 1.04·23-s + 6.83·24-s + 14.0·25-s − 0.495·26-s + 8.09·27-s − 9.76·29-s − 8.32·30-s + 7.10·31-s − 5.70·32-s + ⋯ |
| L(s) = 1 | − 0.455·2-s + 1.70·3-s − 0.792·4-s + 1.95·5-s − 0.778·6-s + 0.817·8-s + 1.91·9-s − 0.890·10-s − 1.40·11-s − 1.35·12-s + 0.213·13-s + 3.33·15-s + 0.419·16-s − 0.0901·17-s − 0.871·18-s + 0.456·19-s − 1.54·20-s + 0.640·22-s + 0.217·23-s + 1.39·24-s + 2.81·25-s − 0.0972·26-s + 1.55·27-s − 1.81·29-s − 1.51·30-s + 1.27·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.015521092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015521092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.644T + 2T^{2} \) |
| 3 | \( 1 - 2.95T + 3T^{2} \) |
| 5 | \( 1 - 4.36T + 5T^{2} \) |
| 11 | \( 1 + 4.65T + 11T^{2} \) |
| 13 | \( 1 - 0.769T + 13T^{2} \) |
| 17 | \( 1 + 0.371T + 17T^{2} \) |
| 19 | \( 1 - 1.98T + 19T^{2} \) |
| 23 | \( 1 - 1.04T + 23T^{2} \) |
| 29 | \( 1 + 9.76T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + 0.873T + 37T^{2} \) |
| 43 | \( 1 - 9.74T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 0.148T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 + 4.27T + 67T^{2} \) |
| 71 | \( 1 + 6.72T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 1.55T + 83T^{2} \) |
| 89 | \( 1 - 2.86T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−9.071850495950276362755886806547, −8.702006985527889203246189691209, −7.80285059873378846346416275515, −7.17921889612405960979493887912, −5.82207838021748360486213761576, −5.19172984872219612832756698105, −4.13276015749814169392492924196, −2.90875531635583785991252123065, −2.29532046657950191934477761073, −1.30425597327974066751385017712,
1.30425597327974066751385017712, 2.29532046657950191934477761073, 2.90875531635583785991252123065, 4.13276015749814169392492924196, 5.19172984872219612832756698105, 5.82207838021748360486213761576, 7.17921889612405960979493887912, 7.80285059873378846346416275515, 8.702006985527889203246189691209, 9.071850495950276362755886806547
