| L(s) = 1 | + 0.00299·2-s + 0.0189·3-s − 1.99·4-s + 1.18·5-s + 5.67e−5·6-s − 5.08·7-s − 0.0119·8-s − 2.99·9-s + 0.00356·10-s + 11-s − 0.0379·12-s + 1.44·13-s − 0.0152·14-s + 0.0225·15-s + 3.99·16-s − 2.75·17-s − 0.00898·18-s − 1.37·19-s − 2.37·20-s − 0.0963·21-s + 0.00299·22-s + 5.74·23-s − 0.000227·24-s − 3.58·25-s + 0.00434·26-s − 0.113·27-s + 10.1·28-s + ⋯ |
| L(s) = 1 | + 0.00211·2-s + 0.0109·3-s − 0.999·4-s + 0.532·5-s + 2.31e−5·6-s − 1.92·7-s − 0.00423·8-s − 0.999·9-s + 0.00112·10-s + 0.301·11-s − 0.0109·12-s + 0.402·13-s − 0.00406·14-s + 0.00582·15-s + 0.999·16-s − 0.668·17-s − 0.00211·18-s − 0.315·19-s − 0.531·20-s − 0.0210·21-s + 0.000638·22-s + 1.19·23-s − 4.63e − 5·24-s − 0.716·25-s + 0.000851·26-s − 0.0218·27-s + 1.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8644067059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8644067059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 0.00299T + 2T^{2} \) |
| 3 | \( 1 - 0.0189T + 3T^{2} \) |
| 5 | \( 1 - 1.18T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 - 3.99T + 29T^{2} \) |
| 31 | \( 1 - 9.63T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 8.25T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 + 4.69T + 53T^{2} \) |
| 59 | \( 1 + 0.0799T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 - 0.191T + 71T^{2} \) |
| 73 | \( 1 - 7.76T + 73T^{2} \) |
| 79 | \( 1 - 4.16T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + 6.45T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−9.351296713962264223770342072439, −8.997585178650006627227276257294, −8.201151434732217752329690716155, −6.83718984318680505801660112006, −6.19787302170507190738770371235, −5.54882412403701463984482398147, −4.38020804657642445798112412144, −3.41057623464467286891418874785, −2.63903250562888796260280743998, −0.64784016423205924349071906109,
0.64784016423205924349071906109, 2.63903250562888796260280743998, 3.41057623464467286891418874785, 4.38020804657642445798112412144, 5.54882412403701463984482398147, 6.19787302170507190738770371235, 6.83718984318680505801660112006, 8.201151434732217752329690716155, 8.997585178650006627227276257294, 9.351296713962264223770342072439
