| L(s) = 1 | − 4·4-s + 16·16-s − 80·29-s − 160·41-s − 98·49-s − 44·61-s − 64·64-s − 320·89-s + 80·101-s − 364·109-s + 320·116-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 640·164-s + 167-s + 238·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 2.75·29-s − 3.90·41-s − 2·49-s − 0.721·61-s − 64-s − 3.59·89-s + 0.792·101-s − 3.33·109-s + 2.75·116-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 3.90·164-s + 0.00598·167-s + 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04869310508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04869310508\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 96 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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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
−10.29728063752796350354402982503, −9.652197036845686763356614594240, −9.376489537846994906992010782212, −8.763785647629164018019312892561, −8.612790887128455351314637716683, −7.927574606019179938259707335583, −7.85039533485365692458881010196, −7.07611940513120865775326939646, −6.80110916338318980829006161127, −6.22014969446764431483932086254, −5.50299938187821274104080290517, −5.41886994382864931517914719396, −4.82791221251137119207238128926, −4.36107934257471320419583540634, −3.61344297103619671108836330955, −3.52004137090482419973307972892, −2.78356309920885525975133623691, −1.72606031449390733016546906619, −1.48799123756900941233887146069, −0.07288480697499949785533325705,
0.07288480697499949785533325705, 1.48799123756900941233887146069, 1.72606031449390733016546906619, 2.78356309920885525975133623691, 3.52004137090482419973307972892, 3.61344297103619671108836330955, 4.36107934257471320419583540634, 4.82791221251137119207238128926, 5.41886994382864931517914719396, 5.50299938187821274104080290517, 6.22014969446764431483932086254, 6.80110916338318980829006161127, 7.07611940513120865775326939646, 7.85039533485365692458881010196, 7.927574606019179938259707335583, 8.612790887128455351314637716683, 8.763785647629164018019312892561, 9.376489537846994906992010782212, 9.652197036845686763356614594240, 10.29728063752796350354402982503
