L(s) = 1 | + 14·19-s − 2·25-s + 10·31-s + 2·37-s + 12·47-s + 12·83-s + 10·103-s + 10·109-s − 12·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 3.21·19-s − 2/5·25-s + 1.79·31-s + 0.328·37-s + 1.75·47-s + 1.31·83-s + 0.985·103-s + 0.957·109-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.357785644\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.357785644\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.040187477939513771311360637997, −7.65611881761990967708807069693, −7.44398135024813468885859670508, −7.26697735051634026350701299758, −6.67815663342761930656247052334, −6.39155363510499330159414877566, −6.01463174764086440807617711324, −5.58821386465766116908128869490, −5.23699617192387628994069607111, −5.13075357967502039251074227677, −4.41389453316476277063711347858, −4.31562701063703723150733037427, −3.66760424636713438528294124816, −3.33429489693943252873393223599, −2.94111092834589446620875486245, −2.64004020004063021094859250249, −2.05546043617351061551266639194, −1.47399982517872016891569393041, −0.871342897843147157232852418468, −0.67987550782813973866333310147,
0.67987550782813973866333310147, 0.871342897843147157232852418468, 1.47399982517872016891569393041, 2.05546043617351061551266639194, 2.64004020004063021094859250249, 2.94111092834589446620875486245, 3.33429489693943252873393223599, 3.66760424636713438528294124816, 4.31562701063703723150733037427, 4.41389453316476277063711347858, 5.13075357967502039251074227677, 5.23699617192387628994069607111, 5.58821386465766116908128869490, 6.01463174764086440807617711324, 6.39155363510499330159414877566, 6.67815663342761930656247052334, 7.26697735051634026350701299758, 7.44398135024813468885859670508, 7.65611881761990967708807069693, 8.040187477939513771311360637997