| L(s) = 1 | − 3·4-s + 8·5-s − 2·9-s + 4·11-s + 4·16-s + 12·19-s − 24·20-s + 38·25-s − 12·29-s + 24·31-s + 6·36-s + 16·41-s − 12·44-s − 16·45-s − 14·49-s + 32·55-s + 4·59-s − 12·61-s − 9·64-s + 4·71-s − 36·76-s + 32·80-s + 9·81-s − 16·89-s + 96·95-s − 8·99-s − 114·100-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 3.57·5-s − 2/3·9-s + 1.20·11-s + 16-s + 2.75·19-s − 5.36·20-s + 38/5·25-s − 2.22·29-s + 4.31·31-s + 36-s + 2.49·41-s − 1.80·44-s − 2.38·45-s − 2·49-s + 4.31·55-s + 0.520·59-s − 1.53·61-s − 9/8·64-s + 0.474·71-s − 4.12·76-s + 3.57·80-s + 81-s − 1.69·89-s + 9.84·95-s − 0.804·99-s − 11.3·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.964277769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.964277769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 50 T^{2} + 651 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 2 T - 67 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.35461599111892442182795793747, −6.88676633638802467167309765587, −6.78983390011302134856340565321, −6.49826836846500585277619445567, −6.27806895542957722015623287653, −6.04858793039101799910238566163, −5.87822973795211060609803679622, −5.72687939116398564035343276555, −5.61483506756877843898924916074, −5.26027619252532551704857829170, −4.90096552373643768927047908376, −4.74208550228399563913522176887, −4.70021068654998012325253263571, −4.43509935186429329320679287075, −3.94571516408082612952941248643, −3.51417291826586896368346270524, −3.39368124992045357126077925450, −2.88212388264555485261955860788, −2.77216192227404656028162942212, −2.62444556002423652189850669084, −1.90068555780013447646161977948, −1.88131942843778828910638235597, −1.34035349048800381065460823844, −0.908918670025635236534274206943, −0.867183432233262071665442271527,
0.867183432233262071665442271527, 0.908918670025635236534274206943, 1.34035349048800381065460823844, 1.88131942843778828910638235597, 1.90068555780013447646161977948, 2.62444556002423652189850669084, 2.77216192227404656028162942212, 2.88212388264555485261955860788, 3.39368124992045357126077925450, 3.51417291826586896368346270524, 3.94571516408082612952941248643, 4.43509935186429329320679287075, 4.70021068654998012325253263571, 4.74208550228399563913522176887, 4.90096552373643768927047908376, 5.26027619252532551704857829170, 5.61483506756877843898924916074, 5.72687939116398564035343276555, 5.87822973795211060609803679622, 6.04858793039101799910238566163, 6.27806895542957722015623287653, 6.49826836846500585277619445567, 6.78983390011302134856340565321, 6.88676633638802467167309765587, 7.35461599111892442182795793747
