| L(s) = 1 | − 1.58·2-s − 3.33·3-s + 0.509·4-s − 0.670·5-s + 5.28·6-s + 1.43·7-s + 2.36·8-s + 8.10·9-s + 1.06·10-s − 4.82·11-s − 1.69·12-s − 2.57·13-s − 2.27·14-s + 2.23·15-s − 4.75·16-s + 17-s − 12.8·18-s − 7.07·19-s − 0.342·20-s − 4.77·21-s + 7.64·22-s − 2.01·23-s − 7.86·24-s − 4.54·25-s + 4.08·26-s − 17.0·27-s + 0.731·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s − 1.92·3-s + 0.254·4-s − 0.300·5-s + 2.15·6-s + 0.541·7-s + 0.834·8-s + 2.70·9-s + 0.336·10-s − 1.45·11-s − 0.490·12-s − 0.714·13-s − 0.607·14-s + 0.577·15-s − 1.18·16-s + 0.242·17-s − 3.02·18-s − 1.62·19-s − 0.0764·20-s − 1.04·21-s + 1.63·22-s − 0.419·23-s − 1.60·24-s − 0.909·25-s + 0.800·26-s − 3.27·27-s + 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1998705779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1998705779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 3 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 0.670T + 5T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 2.01T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 - 4.31T + 31T^{2} \) |
| 37 | \( 1 + 7.72T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 53 | \( 1 + 8.52T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.21T + 83T^{2} \) |
| 89 | \( 1 + 9.93T + 89T^{2} \) |
| 97 | \( 1 + 2.50T + 97T^{2} \) |
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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
−10.26069988653488901840137834197, −9.889862877270294741846852346026, −8.400799434775148022702028879354, −7.75212367541479427076807759383, −6.93123718866202876230989383137, −5.90163981508650179560606255082, −4.88898693639553297529968166777, −4.40432904621793898462371392974, −1.98430509931726855132530595370, −0.46568934378304813524478706072,
0.46568934378304813524478706072, 1.98430509931726855132530595370, 4.40432904621793898462371392974, 4.88898693639553297529968166777, 5.90163981508650179560606255082, 6.93123718866202876230989383137, 7.75212367541479427076807759383, 8.400799434775148022702028879354, 9.889862877270294741846852346026, 10.26069988653488901840137834197
