| L(s) = 1 | − 3-s − 2.49·5-s + 0.917·7-s + 9-s − 3.69·11-s − 5.81·13-s + 2.49·15-s − 0.867·17-s − 6.52·19-s − 0.917·21-s + 4·23-s + 1.23·25-s − 27-s + 7.72·29-s − 2.14·31-s + 3.69·33-s − 2.29·35-s + 2.47·37-s + 5.81·39-s − 9.58·41-s + 9.58·43-s − 2.49·45-s − 1.65·47-s − 6.15·49-s + 0.867·51-s − 3.39·53-s + 9.22·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.11·5-s + 0.346·7-s + 0.333·9-s − 1.11·11-s − 1.61·13-s + 0.644·15-s − 0.210·17-s − 1.49·19-s − 0.200·21-s + 0.834·23-s + 0.246·25-s − 0.192·27-s + 1.43·29-s − 0.385·31-s + 0.643·33-s − 0.387·35-s + 0.406·37-s + 0.930·39-s − 1.49·41-s + 1.46·43-s − 0.372·45-s − 0.241·47-s − 0.879·49-s + 0.121·51-s − 0.466·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5541969719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5541969719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 0.917T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 + 0.867T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 0.0231T + 61T^{2} \) |
| 67 | \( 1 + 5.32T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 8.40T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−8.435082814428826823620358150338, −7.973542726907713664280956517017, −7.21179887226261706941795260078, −6.62885120005099542933085819869, −5.47008631065373589462063235197, −4.74755773192047507406757529398, −4.30075800053290438037065190765, −3.03686469216018985022151986076, −2.13394157552681083533864928902, −0.43964258939683510626910293470,
0.43964258939683510626910293470, 2.13394157552681083533864928902, 3.03686469216018985022151986076, 4.30075800053290438037065190765, 4.74755773192047507406757529398, 5.47008631065373589462063235197, 6.62885120005099542933085819869, 7.21179887226261706941795260078, 7.973542726907713664280956517017, 8.435082814428826823620358150338
