| L(s) = 1 | + 0.465·2-s − 3-s − 1.78·4-s − 0.762·5-s − 0.465·6-s + 2.44·7-s − 1.76·8-s + 9-s − 0.355·10-s + 1.78·12-s + 13-s + 1.14·14-s + 0.762·15-s + 2.74·16-s − 3.36·17-s + 0.465·18-s + 4.15·19-s + 1.35·20-s − 2.44·21-s + 0.825·23-s + 1.76·24-s − 4.41·25-s + 0.465·26-s − 27-s − 4.36·28-s + 1.42·29-s + 0.355·30-s + ⋯ |
| L(s) = 1 | + 0.329·2-s − 0.577·3-s − 0.891·4-s − 0.341·5-s − 0.190·6-s + 0.925·7-s − 0.623·8-s + 0.333·9-s − 0.112·10-s + 0.514·12-s + 0.277·13-s + 0.304·14-s + 0.196·15-s + 0.686·16-s − 0.816·17-s + 0.109·18-s + 0.952·19-s + 0.303·20-s − 0.534·21-s + 0.172·23-s + 0.359·24-s − 0.883·25-s + 0.0913·26-s − 0.192·27-s − 0.824·28-s + 0.264·29-s + 0.0648·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.293290529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.293290529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.465T + 2T^{2} \) |
| 5 | \( 1 + 0.762T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 0.825T + 23T^{2} \) |
| 29 | \( 1 - 1.42T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 0.245T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 + 3.10T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.362719497356877578318051207808, −7.60824477474201115546741573946, −6.82880379242400622998688942416, −5.86623716104047431093237887814, −5.27738476661282068837110714551, −4.59937573363961546801987705697, −4.03286792081237015476158934575, −3.11189240800986441664125816321, −1.76238218591741163120822835466, −0.63693785559797143014026870117,
0.63693785559797143014026870117, 1.76238218591741163120822835466, 3.11189240800986441664125816321, 4.03286792081237015476158934575, 4.59937573363961546801987705697, 5.27738476661282068837110714551, 5.86623716104047431093237887814, 6.82880379242400622998688942416, 7.60824477474201115546741573946, 8.362719497356877578318051207808
