| L(s) = 1 | + 2-s − 5-s − 2·7-s − 8-s − 10-s + 6·11-s − 2·13-s − 2·14-s − 16-s − 6·17-s − 7·19-s + 6·22-s + 9·23-s − 2·26-s − 8·31-s − 6·34-s + 2·35-s + 10·37-s − 7·38-s + 40-s − 3·41-s + 10·43-s + 9·46-s − 11·49-s + 3·53-s − 6·55-s + 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 1.60·19-s + 1.27·22-s + 1.87·23-s − 0.392·26-s − 1.43·31-s − 1.02·34-s + 0.338·35-s + 1.64·37-s − 1.13·38-s + 0.158·40-s − 0.468·41-s + 1.52·43-s + 1.32·46-s − 1.57·49-s + 0.412·53-s − 0.809·55-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.785988349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785988349\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.474248030723274036933124023360, −9.133403481605266037003026980983, −8.862633861183675821626943481547, −8.513056891460069452039407186177, −7.87681680899088239758507011769, −7.46931627856557785847437982474, −6.92243457205358513387163826538, −6.67478561385550140916300161683, −6.28236251870398717486321754110, −6.16420408652214771700532667428, −5.36783902406825406690810595917, −4.81806255509316413449736356982, −4.58221270477119690529292363459, −4.08465295734707262378041037337, −3.66004157959015855203132114914, −3.40884539147433637376264635329, −2.51501277548880281940353046911, −2.28111575988886751847433447476, −1.36433030448518563543418492150, −0.46459806587223489994971596987,
0.46459806587223489994971596987, 1.36433030448518563543418492150, 2.28111575988886751847433447476, 2.51501277548880281940353046911, 3.40884539147433637376264635329, 3.66004157959015855203132114914, 4.08465295734707262378041037337, 4.58221270477119690529292363459, 4.81806255509316413449736356982, 5.36783902406825406690810595917, 6.16420408652214771700532667428, 6.28236251870398717486321754110, 6.67478561385550140916300161683, 6.92243457205358513387163826538, 7.46931627856557785847437982474, 7.87681680899088239758507011769, 8.513056891460069452039407186177, 8.862633861183675821626943481547, 9.133403481605266037003026980983, 9.474248030723274036933124023360
