| L(s) = 1 | + 14·5-s + 22·7-s − 14·9-s + 6·11-s − 34·17-s − 38·19-s + 4·23-s + 97·25-s + 308·35-s − 42·43-s − 196·45-s − 10·47-s + 265·49-s + 84·55-s + 46·61-s − 308·63-s + 78·73-s + 132·77-s + 115·81-s − 12·83-s − 476·85-s − 532·95-s − 84·99-s + 244·101-s + 56·115-s − 748·119-s − 215·121-s + ⋯ |
| L(s) = 1 | + 14/5·5-s + 22/7·7-s − 1.55·9-s + 6/11·11-s − 2·17-s − 2·19-s + 4/23·23-s + 3.87·25-s + 44/5·35-s − 0.976·43-s − 4.35·45-s − 0.212·47-s + 5.40·49-s + 1.52·55-s + 0.754·61-s − 4.88·63-s + 1.06·73-s + 12/7·77-s + 1.41·81-s − 0.144·83-s − 5.59·85-s − 5.59·95-s − 0.848·99-s + 2.41·101-s + 0.486·115-s − 6.28·119-s − 1.77·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.561969201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.561969201\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1794 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 21 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5586 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7410 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3234 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1730 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )( 1 + 94 T + p^{2} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.28308581399942488969058272116, −12.65769332437284218984564154586, −11.63242211949086091319988497813, −11.51755903301418387291056793461, −10.82030881812000784121619400354, −10.76210520633472524788783236008, −10.06223156599613309973975192926, −9.079893014675088962982560744070, −9.020690091650634069050796913326, −8.296702361817719836853141309311, −8.256644258512608096815076929355, −6.99786973484652188153231525181, −6.26116617234065239606643343419, −6.03478965060329443379027960019, −5.12416259033016196125625692446, −5.02443761752915923068876677726, −4.23226180913465932621949627187, −2.34548562951933833460140836405, −2.15736301476957795394401130853, −1.57412450221512034621316246811,
1.57412450221512034621316246811, 2.15736301476957795394401130853, 2.34548562951933833460140836405, 4.23226180913465932621949627187, 5.02443761752915923068876677726, 5.12416259033016196125625692446, 6.03478965060329443379027960019, 6.26116617234065239606643343419, 6.99786973484652188153231525181, 8.256644258512608096815076929355, 8.296702361817719836853141309311, 9.020690091650634069050796913326, 9.079893014675088962982560744070, 10.06223156599613309973975192926, 10.76210520633472524788783236008, 10.82030881812000784121619400354, 11.51755903301418387291056793461, 11.63242211949086091319988497813, 12.65769332437284218984564154586, 13.28308581399942488969058272116
