| L(s) = 1 | − 2.82·5-s − 0.828·7-s + 11-s − 4.82·13-s + 2·17-s − 2·19-s − 2.82·23-s + 3.00·25-s + 3.65·29-s + 9.65·31-s + 2.34·35-s − 7.65·37-s + 0.343·41-s − 0.343·43-s − 12.4·47-s − 6.31·49-s − 6.82·53-s − 2.82·55-s − 1.65·59-s − 3.17·61-s + 13.6·65-s − 11.3·67-s + 4.48·71-s − 13.3·73-s − 0.828·77-s + 4.82·79-s + 9.65·83-s + ⋯ |
| L(s) = 1 | − 1.26·5-s − 0.313·7-s + 0.301·11-s − 1.33·13-s + 0.485·17-s − 0.458·19-s − 0.589·23-s + 0.600·25-s + 0.679·29-s + 1.73·31-s + 0.396·35-s − 1.25·37-s + 0.0535·41-s − 0.0523·43-s − 1.82·47-s − 0.901·49-s − 0.937·53-s − 0.381·55-s − 0.215·59-s − 0.406·61-s + 1.69·65-s − 1.38·67-s + 0.532·71-s − 1.55·73-s − 0.0944·77-s + 0.543·79-s + 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8093440498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8093440498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 6.82T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 4.48T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
| 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
−7.972007944822836248574710012428, −7.45456544310963278037573254931, −6.66916883016553632340557607255, −6.08525708781090043354901226065, −4.77727849978647910191516636017, −4.62605260942949346626473062326, −3.51592438580234671701714218037, −2.99679383975078778404603549766, −1.84033092862646119575103451648, −0.45303433350119744858200838320,
0.45303433350119744858200838320, 1.84033092862646119575103451648, 2.99679383975078778404603549766, 3.51592438580234671701714218037, 4.62605260942949346626473062326, 4.77727849978647910191516636017, 6.08525708781090043354901226065, 6.66916883016553632340557607255, 7.45456544310963278037573254931, 7.972007944822836248574710012428
