L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·8-s + 3·9-s + 4·11-s + 6·12-s + 6·13-s + 5·16-s + 6·17-s + 6·18-s + 8·22-s − 2·23-s + 8·24-s + 12·26-s + 4·27-s + 2·29-s + 2·31-s + 6·32-s + 8·33-s + 12·34-s + 9·36-s + 12·39-s + 6·41-s − 6·43-s + 12·44-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 + 1.20·11-s + 1.73·12-s + 1.66·13-s + 5/4·16-s + 1.45·17-s + 1.41·18-s + 1.70·22-s − 0.417·23-s + 1.63·24-s + 2.35·26-s + 0.769·27-s + 0.371·29-s + 0.359·31-s + 1.06·32-s + 1.39·33-s + 2.05·34-s + 3/2·36-s + 1.92·39-s + 0.937·41-s − 0.914·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(23.29693692\) |
\(L(\frac12)\) |
\(\approx\) |
\(23.29693692\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 93 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 109 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 160 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.996521685721764301114545610317, −7.85216045039248751075841710142, −7.20360453598185968349095782938, −7.07592989225392999374332661947, −6.48453600113197080188617605029, −6.40809705185912567680754865119, −5.85013704648467836338458110191, −5.83916682495327471549301915849, −5.01855497414781134304865680680, −5.01316567326101840885254004335, −4.17636819925127112370933630476, −4.17566598754346814919466813660, −3.63563588888739509978220249553, −3.54899111709051761890063396306, −2.98890483080855565476224587189, −2.83033886182423371836407752989, −1.99672447230822962894557493465, −1.83948679653318268951777883540, −1.08720122017568173208727303114, −0.986981854430028851055716661344,
0.986981854430028851055716661344, 1.08720122017568173208727303114, 1.83948679653318268951777883540, 1.99672447230822962894557493465, 2.83033886182423371836407752989, 2.98890483080855565476224587189, 3.54899111709051761890063396306, 3.63563588888739509978220249553, 4.17566598754346814919466813660, 4.17636819925127112370933630476, 5.01316567326101840885254004335, 5.01855497414781134304865680680, 5.83916682495327471549301915849, 5.85013704648467836338458110191, 6.40809705185912567680754865119, 6.48453600113197080188617605029, 7.07592989225392999374332661947, 7.20360453598185968349095782938, 7.85216045039248751075841710142, 7.996521685721764301114545610317