| L(s) = 1 | − 1.31·2-s − 2.88·3-s − 0.261·4-s − 1.34·5-s + 3.81·6-s + 0.384·7-s + 2.98·8-s + 5.35·9-s + 1.77·10-s − 11-s + 0.756·12-s + 6.11·13-s − 0.507·14-s + 3.88·15-s − 3.40·16-s − 5.49·17-s − 7.05·18-s − 6.99·19-s + 0.351·20-s − 1.11·21-s + 1.31·22-s + 9.47·23-s − 8.61·24-s − 3.19·25-s − 8.06·26-s − 6.79·27-s − 0.100·28-s + ⋯ |
| L(s) = 1 | − 0.932·2-s − 1.66·3-s − 0.130·4-s − 0.600·5-s + 1.55·6-s + 0.145·7-s + 1.05·8-s + 1.78·9-s + 0.560·10-s − 0.301·11-s + 0.218·12-s + 1.69·13-s − 0.135·14-s + 1.00·15-s − 0.852·16-s − 1.33·17-s − 1.66·18-s − 1.60·19-s + 0.0786·20-s − 0.242·21-s + 0.281·22-s + 1.97·23-s − 1.75·24-s − 0.638·25-s − 1.58·26-s − 1.30·27-s − 0.0190·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2893267177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2893267177\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.31T + 2T^{2} \) |
| 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 - 0.384T + 7T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 + 6.99T + 19T^{2} \) |
| 23 | \( 1 - 9.47T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 - 3.96T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 5.98T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 5.83T + 53T^{2} \) |
| 59 | \( 1 - 9.38T + 59T^{2} \) |
| 61 | \( 1 + 0.231T + 61T^{2} \) |
| 67 | \( 1 - 3.76T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 6.85T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 5.51T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−9.562621532225691463884463772332, −8.612130424199721314170114162329, −8.177653590769888570920051887834, −6.85853740091066725722896389480, −6.55303388344443914518389361805, −5.34438259046274931448795149787, −4.60705938119429545982755576717, −3.80541871798114243276636853399, −1.69276377767758832534790435880, −0.49807851986969423083003924246,
0.49807851986969423083003924246, 1.69276377767758832534790435880, 3.80541871798114243276636853399, 4.60705938119429545982755576717, 5.34438259046274931448795149787, 6.55303388344443914518389361805, 6.85853740091066725722896389480, 8.177653590769888570920051887834, 8.612130424199721314170114162329, 9.562621532225691463884463772332
