| L(s) = 1 | + 112·19-s − 250·25-s − 1.04e3·43-s + 286·49-s + 1.76e3·67-s − 2.38e3·73-s + 2.66e3·97-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.35·19-s − 2·25-s − 3.68·43-s + 0.833·49-s + 3.20·67-s − 3.81·73-s + 2.78·97-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 35282 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 89206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 420838 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1190 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 204622 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{2} \) |
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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
−8.318507904020572106412628116361, −8.193451519553105391540441337247, −7.57545181085085858948981377106, −7.45329019166514058826325753139, −6.92980168896278077887958556287, −6.53604766088146059603307039552, −6.10269133089516328102957783207, −5.71277355003209345913494717537, −5.25165198191376392633137159994, −4.98132820141391975109387874418, −4.49310224885488504700031162746, −3.86401364711128171211064469055, −3.49103231842527166934867554437, −3.27675923715750436266686822104, −2.44492913324661168916495789185, −2.13941884836278147849392258665, −1.40970274055540282823523275721, −1.10439143117436727457481903499, 0, 0,
1.10439143117436727457481903499, 1.40970274055540282823523275721, 2.13941884836278147849392258665, 2.44492913324661168916495789185, 3.27675923715750436266686822104, 3.49103231842527166934867554437, 3.86401364711128171211064469055, 4.49310224885488504700031162746, 4.98132820141391975109387874418, 5.25165198191376392633137159994, 5.71277355003209345913494717537, 6.10269133089516328102957783207, 6.53604766088146059603307039552, 6.92980168896278077887958556287, 7.45329019166514058826325753139, 7.57545181085085858948981377106, 8.193451519553105391540441337247, 8.318507904020572106412628116361
