| L(s) = 1 | − 4·4-s − 6·5-s + 16·16-s + 24·20-s + 11·25-s − 84·29-s + 36·41-s − 98·49-s + 44·61-s − 64·64-s − 96·80-s + 156·89-s − 44·100-s + 396·101-s + 364·109-s + 336·116-s + 242·121-s + 84·125-s + 127-s + 131-s + 137-s + 139-s + 504·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s − 6/5·5-s + 16-s + 6/5·20-s + 0.439·25-s − 2.89·29-s + 0.878·41-s − 2·49-s + 0.721·61-s − 64-s − 6/5·80-s + 1.75·89-s − 0.439·100-s + 3.92·101-s + 3.33·109-s + 2.89·116-s + 2·121-s + 0.671·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.47·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6868042810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6868042810\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 130 T + p^{2} T^{2} ) \) |
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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
−12.72008881651602233747332003438, −12.30922974674358471707820234256, −11.47930762886883374758420792930, −11.38404748829505842985063267874, −10.85107819755891534958464792382, −10.00965836455592460324774821131, −9.738080661853311767028994051307, −8.997595708893052525505873262965, −8.753950734827722477203461144494, −8.017103161101761661720012111322, −7.54211510649304447635040624147, −7.32441318783556490779023756060, −6.23887318510839662581091752857, −5.73684749555526150134119994709, −4.92274563950367047607600822537, −4.48491485853067489525421035608, −3.53084878396836873849190959043, −3.52768734533676586455912573460, −1.96445899793537204247253632097, −0.50556913704680698130421851411,
0.50556913704680698130421851411, 1.96445899793537204247253632097, 3.52768734533676586455912573460, 3.53084878396836873849190959043, 4.48491485853067489525421035608, 4.92274563950367047607600822537, 5.73684749555526150134119994709, 6.23887318510839662581091752857, 7.32441318783556490779023756060, 7.54211510649304447635040624147, 8.017103161101761661720012111322, 8.753950734827722477203461144494, 8.997595708893052525505873262965, 9.738080661853311767028994051307, 10.00965836455592460324774821131, 10.85107819755891534958464792382, 11.38404748829505842985063267874, 11.47930762886883374758420792930, 12.30922974674358471707820234256, 12.72008881651602233747332003438
