| L(s) = 1 | − 2-s − 0.167·3-s + 4-s − 1.86·5-s + 0.167·6-s − 4.61·7-s − 8-s − 2.97·9-s + 1.86·10-s + 11-s − 0.167·12-s + 0.766·13-s + 4.61·14-s + 0.310·15-s + 16-s + 3.09·17-s + 2.97·18-s + 2.62·19-s − 1.86·20-s + 0.771·21-s − 22-s + 7.16·23-s + 0.167·24-s − 1.53·25-s − 0.766·26-s + 0.997·27-s − 4.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0964·3-s + 0.5·4-s − 0.832·5-s + 0.0682·6-s − 1.74·7-s − 0.353·8-s − 0.990·9-s + 0.588·10-s + 0.301·11-s − 0.0482·12-s + 0.212·13-s + 1.23·14-s + 0.0802·15-s + 0.250·16-s + 0.750·17-s + 0.700·18-s + 0.602·19-s − 0.416·20-s + 0.168·21-s − 0.213·22-s + 1.49·23-s + 0.0341·24-s − 0.307·25-s − 0.150·26-s + 0.192·27-s − 0.872·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 0.167T + 3T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 13 | \( 1 - 0.766T + 13T^{2} \) |
| 17 | \( 1 - 3.09T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 + 0.0691T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 - 0.348T + 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 - 0.00342T + 47T^{2} \) |
| 53 | \( 1 + 8.61T + 53T^{2} \) |
| 59 | \( 1 - 4.25T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 2.65T + 79T^{2} \) |
| 83 | \( 1 - 0.0519T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
| 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
−8.050934707453285958760188075383, −7.31725204794549322123267687982, −6.66920943568344859303677434379, −5.98417912543008910271923275968, −5.25232246550564230619511300856, −3.87222840939719657878861972366, −3.27238980574656494123475940015, −2.66013942677836885610899161573, −0.977580836695719589652754714221, 0,
0.977580836695719589652754714221, 2.66013942677836885610899161573, 3.27238980574656494123475940015, 3.87222840939719657878861972366, 5.25232246550564230619511300856, 5.98417912543008910271923275968, 6.66920943568344859303677434379, 7.31725204794549322123267687982, 8.050934707453285958760188075383
