| L(s) = 1 | − 4-s + 2·7-s − 9-s + 6·11-s + 12·13-s + 16-s + 10·25-s − 2·28-s − 10·29-s + 36-s + 4·37-s − 4·43-s − 6·44-s − 20·47-s − 11·49-s − 12·52-s − 14·53-s + 8·59-s − 2·63-s − 64-s + 12·77-s + 81-s − 4·89-s + 24·91-s − 6·97-s − 6·99-s − 10·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s − 1/3·9-s + 1.80·11-s + 3.32·13-s + 1/4·16-s + 2·25-s − 0.377·28-s − 1.85·29-s + 1/6·36-s + 0.657·37-s − 0.609·43-s − 0.904·44-s − 2.91·47-s − 1.57·49-s − 1.66·52-s − 1.92·53-s + 1.04·59-s − 0.251·63-s − 1/8·64-s + 1.36·77-s + 1/9·81-s − 0.423·89-s + 2.51·91-s − 0.609·97-s − 0.603·99-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909857437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909857437\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 53 | $C_2$ | \( 1 + 14 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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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
−11.57242657276930132609566913615, −11.32240948749244994987787156489, −11.01021709918580981822077551102, −10.89302439666954869711785416087, −9.826492543999002807719205358380, −9.498546148869967035699929292908, −8.895154503823129703888060814011, −8.668100040311510014424540758722, −8.212984698160145099802781242731, −7.87434568459450798911016170792, −6.71113258157996482042208227185, −6.59933204672422019229421069169, −6.08906756361666716039821253485, −5.45687261131433530482947320616, −4.76897564156826014581476349491, −4.18702663207471237505370143731, −3.44145095907490100393639043538, −3.36021797288687143404927431267, −1.50777055975745637803259303139, −1.36125716809670900919948226127,
1.36125716809670900919948226127, 1.50777055975745637803259303139, 3.36021797288687143404927431267, 3.44145095907490100393639043538, 4.18702663207471237505370143731, 4.76897564156826014581476349491, 5.45687261131433530482947320616, 6.08906756361666716039821253485, 6.59933204672422019229421069169, 6.71113258157996482042208227185, 7.87434568459450798911016170792, 8.212984698160145099802781242731, 8.668100040311510014424540758722, 8.895154503823129703888060814011, 9.498546148869967035699929292908, 9.826492543999002807719205358380, 10.89302439666954869711785416087, 11.01021709918580981822077551102, 11.32240948749244994987787156489, 11.57242657276930132609566913615
