| L(s) = 1 | − 0.369·2-s + 3-s − 1.86·4-s − 5-s − 0.369·6-s − 1.65·7-s + 1.42·8-s + 9-s + 0.369·10-s + 1.81·11-s − 1.86·12-s + 1.97·13-s + 0.611·14-s − 15-s + 3.20·16-s + 6.89·17-s − 0.369·18-s + 6.46·19-s + 1.86·20-s − 1.65·21-s − 0.671·22-s − 7.75·23-s + 1.42·24-s + 25-s − 0.730·26-s + 27-s + 3.08·28-s + ⋯ |
| L(s) = 1 | − 0.261·2-s + 0.577·3-s − 0.931·4-s − 0.447·5-s − 0.150·6-s − 0.625·7-s + 0.504·8-s + 0.333·9-s + 0.116·10-s + 0.548·11-s − 0.538·12-s + 0.548·13-s + 0.163·14-s − 0.258·15-s + 0.800·16-s + 1.67·17-s − 0.0870·18-s + 1.48·19-s + 0.416·20-s − 0.361·21-s − 0.143·22-s − 1.61·23-s + 0.291·24-s + 0.200·25-s − 0.143·26-s + 0.192·27-s + 0.582·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.565427826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565427826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.369T + 2T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 + 7.75T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 + 0.807T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 4.70T + 61T^{2} \) |
| 67 | \( 1 - 1.01T + 67T^{2} \) |
| 71 | \( 1 + 1.59T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 + 5.88T + 97T^{2} \) |
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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
−8.077642580432221671273129318186, −7.65550196289067974364334960560, −6.83562392103984813917663605420, −5.83606596160537044482157052916, −5.24807144749070051802169214641, −4.18473251113041391750726906037, −3.58753769320938136492208308371, −3.14114173914827395355229736905, −1.62507262301195015508147985197, −0.71256444933929877565783880882,
0.71256444933929877565783880882, 1.62507262301195015508147985197, 3.14114173914827395355229736905, 3.58753769320938136492208308371, 4.18473251113041391750726906037, 5.24807144749070051802169214641, 5.83606596160537044482157052916, 6.83562392103984813917663605420, 7.65550196289067974364334960560, 8.077642580432221671273129318186
