| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·8-s + 3·9-s + 6·12-s − 4·13-s + 5·16-s + 2·17-s + 6·18-s + 8·19-s + 8·23-s + 8·24-s − 8·26-s + 4·27-s + 12·29-s + 8·31-s + 6·32-s + 4·34-s + 9·36-s − 12·37-s + 16·38-s − 8·39-s + 4·41-s − 8·43-s + 16·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.41·8-s + 9-s + 1.73·12-s − 1.10·13-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s + 1.66·23-s + 1.63·24-s − 1.56·26-s + 0.769·27-s + 2.22·29-s + 1.43·31-s + 1.06·32-s + 0.685·34-s + 3/2·36-s − 1.97·37-s + 2.59·38-s − 1.28·39-s + 0.624·41-s − 1.21·43-s + 2.35·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.57530258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.57530258\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.082550147459862918676075951706, −8.606682605643484209080879198283, −8.127916647169046251029335368726, −8.062299306457441686802596270243, −7.31325713442745415679473846396, −7.18828732564724250752809734899, −6.84895881498179847375877778963, −6.50667932303110474761670697883, −5.94152664604761259414837900025, −5.21802591240586651643094799364, −5.09674292455891256716034926920, −4.90474911446282460543269684986, −4.28826888197865574242287274394, −3.81978772612130592958545511318, −3.22895280552767533646838242355, −3.08535406564464226280358519328, −2.60829502822413148291791873420, −2.28706944761092575345808084901, −1.31410150339493194386988013659, −1.01371369103375821978881681616,
1.01371369103375821978881681616, 1.31410150339493194386988013659, 2.28706944761092575345808084901, 2.60829502822413148291791873420, 3.08535406564464226280358519328, 3.22895280552767533646838242355, 3.81978772612130592958545511318, 4.28826888197865574242287274394, 4.90474911446282460543269684986, 5.09674292455891256716034926920, 5.21802591240586651643094799364, 5.94152664604761259414837900025, 6.50667932303110474761670697883, 6.84895881498179847375877778963, 7.18828732564724250752809734899, 7.31325713442745415679473846396, 8.062299306457441686802596270243, 8.127916647169046251029335368726, 8.606682605643484209080879198283, 9.082550147459862918676075951706
