| L(s) = 1 | − 6.61·2-s + 9.41·3-s + 11.7·4-s − 25·5-s − 62.2·6-s − 82.7·7-s + 133.·8-s − 154.·9-s + 165.·10-s − 121·11-s + 110.·12-s + 261.·13-s + 547.·14-s − 235.·15-s − 1.26e3·16-s + 1.32e3·17-s + 1.02e3·18-s + 361·19-s − 294.·20-s − 778.·21-s + 800.·22-s + 3.17e3·23-s + 1.25e3·24-s + 625·25-s − 1.72e3·26-s − 3.74e3·27-s − 974.·28-s + ⋯ |
| L(s) = 1 | − 1.16·2-s + 0.603·3-s + 0.367·4-s − 0.447·5-s − 0.706·6-s − 0.638·7-s + 0.739·8-s − 0.635·9-s + 0.523·10-s − 0.301·11-s + 0.222·12-s + 0.428·13-s + 0.746·14-s − 0.269·15-s − 1.23·16-s + 1.10·17-s + 0.743·18-s + 0.229·19-s − 0.164·20-s − 0.385·21-s + 0.352·22-s + 1.25·23-s + 0.446·24-s + 0.200·25-s − 0.501·26-s − 0.987·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7554693149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7554693149\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 6.61T + 32T^{2} \) |
| 3 | \( 1 - 9.41T + 243T^{2} \) |
| 7 | \( 1 + 82.7T + 1.68e4T^{2} \) |
| 13 | \( 1 - 261.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 105.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 300.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.63e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.61e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.72e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.41e4T + 8.58e9T^{2} \) |
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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
−9.224110632175494565355238405930, −8.414267405294266775023910888297, −7.79812035235985219231520261591, −7.10649122570453717691011904488, −5.93517033467450980131942060660, −4.85149784834598250110551398165, −3.55924731497341462342547421921, −2.88284182702748829912859138506, −1.51847781102999373574868703602, −0.45281484649887713843065096908,
0.45281484649887713843065096908, 1.51847781102999373574868703602, 2.88284182702748829912859138506, 3.55924731497341462342547421921, 4.85149784834598250110551398165, 5.93517033467450980131942060660, 7.10649122570453717691011904488, 7.79812035235985219231520261591, 8.414267405294266775023910888297, 9.224110632175494565355238405930
