| L(s) = 1 | + 4-s + 2·9-s + 11-s + 16-s + 6·25-s + 2·36-s + 4·37-s + 44-s − 7·49-s + 4·53-s + 64-s + 24·67-s − 8·71-s − 5·81-s + 2·99-s + 6·100-s − 20·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 4·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2/3·9-s + 0.301·11-s + 1/4·16-s + 6/5·25-s + 1/3·36-s + 0.657·37-s + 0.150·44-s − 49-s + 0.549·53-s + 1/8·64-s + 2.93·67-s − 0.949·71-s − 5/9·81-s + 0.201·99-s + 3/5·100-s − 1.88·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.240779327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240779327\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.901043807505772270496559580452, −8.395607811720954841765968672470, −8.011119657909147300953524820841, −7.43887961932470076210113809368, −6.92729386640687152635053717930, −6.68958429938545382809359677306, −6.13457959830705833057754257493, −5.52075582489292781309550153862, −4.99724884061860764884236484895, −4.41724586149363676215234692744, −3.87528143167997590960012891784, −3.20415769548060747117531297053, −2.57328165327186777306749670782, −1.79727523088765617712474080432, −0.960769318426187613156128602249,
0.960769318426187613156128602249, 1.79727523088765617712474080432, 2.57328165327186777306749670782, 3.20415769548060747117531297053, 3.87528143167997590960012891784, 4.41724586149363676215234692744, 4.99724884061860764884236484895, 5.52075582489292781309550153862, 6.13457959830705833057754257493, 6.68958429938545382809359677306, 6.92729386640687152635053717930, 7.43887961932470076210113809368, 8.011119657909147300953524820841, 8.395607811720954841765968672470, 8.901043807505772270496559580452
