| L(s) = 1 | − 4·3-s + 4-s − 4·5-s + 6·9-s − 2·11-s − 4·12-s + 16·15-s + 16-s − 4·20-s − 16·23-s + 2·25-s + 4·27-s − 16·31-s + 8·33-s + 6·36-s + 4·37-s − 2·44-s − 24·45-s + 8·47-s − 4·48-s + 2·49-s − 8·53-s + 8·55-s + 16·59-s + 16·60-s + 64-s − 4·67-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s − 1.78·5-s + 2·9-s − 0.603·11-s − 1.15·12-s + 4.13·15-s + 1/4·16-s − 0.894·20-s − 3.33·23-s + 2/5·25-s + 0.769·27-s − 2.87·31-s + 1.39·33-s + 36-s + 0.657·37-s − 0.301·44-s − 3.57·45-s + 1.16·47-s − 0.577·48-s + 2/7·49-s − 1.09·53-s + 1.07·55-s + 2.08·59-s + 2.06·60-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 813604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 813604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.024357451025613311326857967234, −7.50501018771004978536853111924, −7.04169032820714443488725174993, −6.70748925180204914131686110030, −5.94236815873722182165691595935, −5.69856539643665466176583402943, −5.64804010098781556115362461940, −4.86924096907948566839386350025, −4.29503464325844101707308794985, −3.86994574951596423396651108733, −3.54075497823589975288034605375, −2.49154151345673781514406094737, −1.81218327277471613539827363635, −0.51007563820159210654692309037, 0,
0.51007563820159210654692309037, 1.81218327277471613539827363635, 2.49154151345673781514406094737, 3.54075497823589975288034605375, 3.86994574951596423396651108733, 4.29503464325844101707308794985, 4.86924096907948566839386350025, 5.64804010098781556115362461940, 5.69856539643665466176583402943, 5.94236815873722182165691595935, 6.70748925180204914131686110030, 7.04169032820714443488725174993, 7.50501018771004978536853111924, 8.024357451025613311326857967234
