L(s) = 1 | + 1.70·2-s + 3-s + 0.916·4-s − 1.21·5-s + 1.70·6-s − 2.14·7-s − 1.85·8-s + 9-s − 2.08·10-s + 0.285·11-s + 0.916·12-s + 3.50·13-s − 3.65·14-s − 1.21·15-s − 4.99·16-s − 17-s + 1.70·18-s + 5.93·19-s − 1.11·20-s − 2.14·21-s + 0.487·22-s + 6.24·23-s − 1.85·24-s − 3.51·25-s + 5.99·26-s + 27-s − 1.96·28-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 0.577·3-s + 0.458·4-s − 0.545·5-s + 0.697·6-s − 0.808·7-s − 0.654·8-s + 0.333·9-s − 0.658·10-s + 0.0861·11-s + 0.264·12-s + 0.973·13-s − 0.976·14-s − 0.314·15-s − 1.24·16-s − 0.242·17-s + 0.402·18-s + 1.36·19-s − 0.249·20-s − 0.467·21-s + 0.104·22-s + 1.30·23-s − 0.377·24-s − 0.702·25-s + 1.17·26-s + 0.192·27-s − 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.550824707\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.550824707\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 0.285T + 11T^{2} \) |
| 13 | \( 1 - 3.50T + 13T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 7.18T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 - 8.52T + 53T^{2} \) |
| 59 | \( 1 - 8.76T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 9.27T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 - 4.13T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79535555144664027203340700483, −6.80712394102511735341375222015, −6.59331821432200539818512718578, −5.40817711020503953812912703044, −5.17886272357501156331897154184, −3.88078324100413411318548832473, −3.70394468350510372407412703897, −3.10884149902647771011360005228, −2.13866345417666347904426632400, −0.74392944517711457273215052246,
0.74392944517711457273215052246, 2.13866345417666347904426632400, 3.10884149902647771011360005228, 3.70394468350510372407412703897, 3.88078324100413411318548832473, 5.17886272357501156331897154184, 5.40817711020503953812912703044, 6.59331821432200539818512718578, 6.80712394102511735341375222015, 7.79535555144664027203340700483