| L(s) = 1 | − 2.88·2-s − 2.89·3-s + 0.299·4-s − 5·5-s + 8.32·6-s + 22.1·8-s − 18.6·9-s + 14.4·10-s − 46.4·11-s − 0.865·12-s − 31.0·13-s + 14.4·15-s − 66.3·16-s − 61.8·17-s + 53.7·18-s + 24.6·19-s − 1.49·20-s + 133.·22-s − 154.·23-s − 64.1·24-s + 25·25-s + 89.4·26-s + 131.·27-s + 200.·29-s − 41.6·30-s + 129.·31-s + 13.5·32-s + ⋯ |
| L(s) = 1 | − 1.01·2-s − 0.556·3-s + 0.0374·4-s − 0.447·5-s + 0.566·6-s + 0.980·8-s − 0.690·9-s + 0.455·10-s − 1.27·11-s − 0.0208·12-s − 0.662·13-s + 0.248·15-s − 1.03·16-s − 0.882·17-s + 0.703·18-s + 0.297·19-s − 0.0167·20-s + 1.29·22-s − 1.40·23-s − 0.545·24-s + 0.200·25-s + 0.674·26-s + 0.940·27-s + 1.28·29-s − 0.253·30-s + 0.748·31-s + 0.0748·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3061228713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3061228713\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.88T + 8T^{2} \) |
| 3 | \( 1 + 2.89T + 27T^{2} \) |
| 11 | \( 1 + 46.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 77.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 169.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 696.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 2.33T + 3.00e5T^{2} \) |
| 71 | \( 1 + 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 752.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 842.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 23.4T + 9.12e5T^{2} \) |
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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
−11.43165200274472700017761892884, −10.53014247780818219498374251537, −9.854211577476550482087643315580, −8.549651174073281269567000457633, −8.008833160785747089668256380311, −6.87332979830539602552055633382, −5.44783445946901084243456785488, −4.42665986229446890989578114942, −2.51194248576325288467810013896, −0.45982755472356528798824533563,
0.45982755472356528798824533563, 2.51194248576325288467810013896, 4.42665986229446890989578114942, 5.44783445946901084243456785488, 6.87332979830539602552055633382, 8.008833160785747089668256380311, 8.549651174073281269567000457633, 9.854211577476550482087643315580, 10.53014247780818219498374251537, 11.43165200274472700017761892884
