L(s) = 1 | + 0.602·3-s + 5-s + 7-s − 2.63·9-s − 5.63·11-s − 4.43·13-s + 0.602·15-s − 7.46·17-s + 1.20·19-s + 0.602·21-s + 7.27·23-s + 25-s − 3.39·27-s + 6.67·29-s + 10.0·31-s − 3.39·33-s + 35-s − 4.86·37-s − 2.67·39-s + 4.79·41-s − 2·43-s − 2.63·45-s − 6.43·47-s + 49-s − 4.49·51-s + 2.06·53-s − 5.63·55-s + ⋯ |
L(s) = 1 | + 0.347·3-s + 0.447·5-s + 0.377·7-s − 0.878·9-s − 1.69·11-s − 1.22·13-s + 0.155·15-s − 1.81·17-s + 0.276·19-s + 0.131·21-s + 1.51·23-s + 0.200·25-s − 0.653·27-s + 1.23·29-s + 1.80·31-s − 0.591·33-s + 0.169·35-s − 0.799·37-s − 0.427·39-s + 0.748·41-s − 0.304·43-s − 0.393·45-s − 0.938·47-s + 0.142·49-s − 0.630·51-s + 0.284·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.554390713\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.554390713\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 0.602T + 3T^{2} \) |
| 11 | \( 1 + 5.63T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4.86T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.43T + 47T^{2} \) |
| 53 | \( 1 - 2.06T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 + 1.27T + 61T^{2} \) |
| 67 | \( 1 - 7.20T + 67T^{2} \) |
| 71 | \( 1 + 6.79T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.83173712943194211909410641994, −7.07543776625412678568360754193, −6.45623979125198782709378476587, −5.53338269715418075358736166768, −4.86849927874125337335634459279, −4.59733889458508275539930531427, −3.03186786780264548231439837695, −2.68437449840515177758023345456, −2.05980786005600724203142224978, −0.55411033651999078661338626877,
0.55411033651999078661338626877, 2.05980786005600724203142224978, 2.68437449840515177758023345456, 3.03186786780264548231439837695, 4.59733889458508275539930531427, 4.86849927874125337335634459279, 5.53338269715418075358736166768, 6.45623979125198782709378476587, 7.07543776625412678568360754193, 7.83173712943194211909410641994