| L(s) = 1 | + 4.12·2-s − 254.·3-s − 494.·4-s + 151.·5-s − 1.04e3·6-s + 9.40e3·7-s − 4.15e3·8-s + 4.48e4·9-s + 625.·10-s − 5.69e4·11-s + 1.25e5·12-s − 6.08e4·13-s + 3.88e4·14-s − 3.85e4·15-s + 2.36e5·16-s + 8.35e4·17-s + 1.85e5·18-s + 1.00e6·19-s − 7.50e4·20-s − 2.39e6·21-s − 2.35e5·22-s + 1.35e6·23-s + 1.05e6·24-s − 1.93e6·25-s − 2.51e5·26-s − 6.39e6·27-s − 4.65e6·28-s + ⋯ |
| L(s) = 1 | + 0.182·2-s − 1.81·3-s − 0.966·4-s + 0.108·5-s − 0.330·6-s + 1.48·7-s − 0.358·8-s + 2.27·9-s + 0.0197·10-s − 1.17·11-s + 1.75·12-s − 0.591·13-s + 0.270·14-s − 0.196·15-s + 0.901·16-s + 0.242·17-s + 0.416·18-s + 1.77·19-s − 0.104·20-s − 2.68·21-s − 0.214·22-s + 1.01·23-s + 0.650·24-s − 0.988·25-s − 0.107·26-s − 2.31·27-s − 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8422762602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8422762602\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - 8.35e4T \) |
| good | 2 | \( 1 - 4.12T + 512T^{2} \) |
| 3 | \( 1 + 254.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 151.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.69e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.08e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.00e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.35e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.81e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.09e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.14e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.87e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.32e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.80e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.49e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.24e9T + 7.60e17T^{2} \) |
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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
−17.17096707864571088822781875455, −15.58059279670676830910762326962, −13.88180122252500421742811551628, −12.40966610854059651971919210545, −11.30396254096535651521154351679, −9.987133772954341697833874929165, −7.70653694709031935913070747997, −5.41543927693687106766573064045, −4.82801047484950600224407818281, −0.849171790121337493242783066568,
0.849171790121337493242783066568, 4.82801047484950600224407818281, 5.41543927693687106766573064045, 7.70653694709031935913070747997, 9.987133772954341697833874929165, 11.30396254096535651521154351679, 12.40966610854059651971919210545, 13.88180122252500421742811551628, 15.58059279670676830910762326962, 17.17096707864571088822781875455
