| L(s) = 1 | − 2.74·2-s − 2.41·3-s + 5.52·4-s + 0.624·5-s + 6.61·6-s − 9.65·8-s + 2.82·9-s − 1.71·10-s + 2.46·11-s − 13.3·12-s − 1.50·15-s + 15.4·16-s + 0.0180·17-s − 7.74·18-s + 5.34·19-s + 3.44·20-s − 6.76·22-s − 3.32·23-s + 23.3·24-s − 4.61·25-s + 0.422·27-s + 4.10·29-s + 4.13·30-s − 5.16·31-s − 23.0·32-s − 5.95·33-s − 0.0495·34-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 1.39·3-s + 2.76·4-s + 0.279·5-s + 2.70·6-s − 3.41·8-s + 0.941·9-s − 0.541·10-s + 0.743·11-s − 3.84·12-s − 0.389·15-s + 3.86·16-s + 0.00438·17-s − 1.82·18-s + 1.22·19-s + 0.770·20-s − 1.44·22-s − 0.692·23-s + 4.75·24-s − 0.922·25-s + 0.0812·27-s + 0.762·29-s + 0.754·30-s − 0.927·31-s − 4.07·32-s − 1.03·33-s − 0.00849·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4835810895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4835810895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.624T + 5T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 17 | \( 1 - 0.0180T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 - 4.10T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 - 0.158T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 8.51T + 59T^{2} \) |
| 61 | \( 1 + 9.72T + 61T^{2} \) |
| 67 | \( 1 - 0.0411T + 67T^{2} \) |
| 71 | \( 1 - 1.95T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 8.81T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 - 5.04T + 97T^{2} \) |
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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
−7.74818120805344377963416040009, −7.22513244747343047007205500046, −6.56169740611053244966514759813, −5.88316975533359095982694823344, −5.60852645395737511048699740306, −4.33525324245285856497232503719, −3.19520488742888046132881811482, −2.15177986495443199708957263989, −1.26633629081205555630418091233, −0.56151766839423899512720254694,
0.56151766839423899512720254694, 1.26633629081205555630418091233, 2.15177986495443199708957263989, 3.19520488742888046132881811482, 4.33525324245285856497232503719, 5.60852645395737511048699740306, 5.88316975533359095982694823344, 6.56169740611053244966514759813, 7.22513244747343047007205500046, 7.74818120805344377963416040009
