| L(s) = 1 | − 4·5-s − 6·13-s + 8·17-s + 5·25-s + 4·29-s − 2·37-s + 16·41-s − 14·49-s − 8·53-s − 10·61-s + 24·65-s − 6·73-s − 32·85-s + 16·89-s − 36·97-s − 40·101-s − 6·109-s + 16·113-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.66·13-s + 1.94·17-s + 25-s + 0.742·29-s − 0.328·37-s + 2.49·41-s − 2·49-s − 1.09·53-s − 1.28·61-s + 2.97·65-s − 0.702·73-s − 3.47·85-s + 1.69·89-s − 3.65·97-s − 3.98·101-s − 0.574·109-s + 1.50·113-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−7.85248809531755222527953073107, −7.80943779052706560209886633236, −7.38924459704379828540474071132, −7.21360677544327579626231383455, −6.59394465722425304341457255592, −6.41374109088574289489989260190, −5.63394505049767488379471453001, −5.59410678323158743238873034474, −5.02706023483313582911087032764, −4.58346329669266778116916996326, −4.38027692872652530015933425313, −3.97200775638225143695706534016, −3.43868104002429363539047202533, −3.16379874924908033649163286040, −2.72355090716680571135290985428, −2.33505517931353462318356823933, −1.39666520891756105457675568358, −1.09925122458770635824964610866, 0, 0,
1.09925122458770635824964610866, 1.39666520891756105457675568358, 2.33505517931353462318356823933, 2.72355090716680571135290985428, 3.16379874924908033649163286040, 3.43868104002429363539047202533, 3.97200775638225143695706534016, 4.38027692872652530015933425313, 4.58346329669266778116916996326, 5.02706023483313582911087032764, 5.59410678323158743238873034474, 5.63394505049767488379471453001, 6.41374109088574289489989260190, 6.59394465722425304341457255592, 7.21360677544327579626231383455, 7.38924459704379828540474071132, 7.80943779052706560209886633236, 7.85248809531755222527953073107
