| L(s) = 1 | − 4-s + 5·9-s + 2·11-s + 16-s + 14·19-s + 6·29-s − 14·31-s − 5·36-s + 12·41-s − 2·44-s − 11·49-s + 12·59-s − 2·61-s − 64-s + 6·71-s − 14·76-s + 20·79-s + 16·81-s − 18·89-s + 10·99-s − 12·101-s − 28·109-s − 6·116-s + 3·121-s + 14·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 5/3·9-s + 0.603·11-s + 1/4·16-s + 3.21·19-s + 1.11·29-s − 2.51·31-s − 5/6·36-s + 1.87·41-s − 0.301·44-s − 1.57·49-s + 1.56·59-s − 0.256·61-s − 1/8·64-s + 0.712·71-s − 1.60·76-s + 2.25·79-s + 16/9·81-s − 1.90·89-s + 1.00·99-s − 1.19·101-s − 2.68·109-s − 0.557·116-s + 3/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.177429909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.177429909\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−10.89529392920413726058338221080, −10.62634845114150992056375287592, −9.872573565846846921262081192773, −9.670757778680521386357921887007, −9.293935604123620728143844931281, −9.181249842695240825571696682267, −8.160271807237518723995401849657, −7.943522986215866105661776170970, −7.25649580919878378594254372166, −7.15989353767523121027878368847, −6.61924093550197951839761950528, −5.86337278874472170410394138536, −5.30824569154342036016761879281, −5.02102803993561741602291531826, −4.33981681651439001046638782509, −3.73528919509568446705559227717, −3.46807096328563860330953783621, −2.55311686528491143720012134224, −1.46813473416451137846983400291, −1.03521147724708745782887219622,
1.03521147724708745782887219622, 1.46813473416451137846983400291, 2.55311686528491143720012134224, 3.46807096328563860330953783621, 3.73528919509568446705559227717, 4.33981681651439001046638782509, 5.02102803993561741602291531826, 5.30824569154342036016761879281, 5.86337278874472170410394138536, 6.61924093550197951839761950528, 7.15989353767523121027878368847, 7.25649580919878378594254372166, 7.943522986215866105661776170970, 8.160271807237518723995401849657, 9.181249842695240825571696682267, 9.293935604123620728143844931281, 9.670757778680521386357921887007, 9.872573565846846921262081192773, 10.62634845114150992056375287592, 10.89529392920413726058338221080
