| L(s) = 1 | + 3·3-s + 2·5-s + 22·7-s + 6·9-s − 16·11-s − 9·13-s + 6·15-s + 26·17-s + 38·19-s + 66·21-s + 32·23-s + 25·25-s + 9·27-s − 42·29-s − 48·33-s + 44·35-s − 27·39-s − 24·41-s − 47·43-s + 12·45-s − 70·47-s + 265·49-s + 78·51-s − 12·53-s − 32·55-s + 114·57-s − 186·59-s + ⋯ |
| L(s) = 1 | + 3-s + 2/5·5-s + 22/7·7-s + 2/3·9-s − 1.45·11-s − 0.692·13-s + 2/5·15-s + 1.52·17-s + 2·19-s + 22/7·21-s + 1.39·23-s + 25-s + 1/3·27-s − 1.44·29-s − 1.45·33-s + 1.25·35-s − 0.692·39-s − 0.585·41-s − 1.09·43-s + 4/15·45-s − 1.48·47-s + 5.40·49-s + 1.52·51-s − 0.226·53-s − 0.581·55-s + 2·57-s − 3.15·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.586065036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.586065036\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - 21 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 9 T + 196 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T + 387 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 32 T + 495 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 42 T + 1429 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 551 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 24 T + 1873 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 47 T + 360 T^{2} + 47 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 70 T + 2691 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 2857 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 186 T + 15013 T^{2} + 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 35 T - 2496 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 27 T + 4732 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 228 T + 22369 T^{2} + 228 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 59 T - 1848 T^{2} + 59 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 45 T + 6916 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 138 T + 14269 T^{2} + 138 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 36 T + 9841 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.07958911426241848436883711775, −11.73313142372364817496351066092, −11.24204681568829726309178418410, −10.85849177646691704159787393974, −10.26436039208366408612323075916, −9.885552195682223847904935860243, −9.156888197612863317537607017018, −8.779415596092775095483424139989, −8.016127382167486194226238549458, −7.912835448168902647937302888879, −7.42232822159670474359408117027, −7.20592402631224009994469406772, −5.69709142786819771428670552392, −5.30715707017173930844420518473, −4.87341828925758839719528678422, −4.59809724882922797611603342844, −3.12640719290374565899461313628, −2.94408852936301464917364552226, −1.55406465835299107130054977578, −1.54056753686115807577553307118,
1.54056753686115807577553307118, 1.55406465835299107130054977578, 2.94408852936301464917364552226, 3.12640719290374565899461313628, 4.59809724882922797611603342844, 4.87341828925758839719528678422, 5.30715707017173930844420518473, 5.69709142786819771428670552392, 7.20592402631224009994469406772, 7.42232822159670474359408117027, 7.912835448168902647937302888879, 8.016127382167486194226238549458, 8.779415596092775095483424139989, 9.156888197612863317537607017018, 9.885552195682223847904935860243, 10.26436039208366408612323075916, 10.85849177646691704159787393974, 11.24204681568829726309178418410, 11.73313142372364817496351066092, 12.07958911426241848436883711775
