| L(s) = 1 | − 72·11-s + 38·13-s − 216·23-s − 182·25-s − 218·37-s − 936·47-s − 181·49-s + 72·59-s − 290·61-s − 1.15e3·71-s − 1.61e3·73-s + 720·83-s + 1.91e3·97-s − 3.09e3·107-s + 428·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s − 2.73e3·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.31e3·169-s + ⋯ |
| L(s) = 1 | − 1.97·11-s + 0.810·13-s − 1.95·23-s − 1.45·25-s − 0.968·37-s − 2.90·47-s − 0.527·49-s + 0.158·59-s − 0.608·61-s − 1.92·71-s − 2.59·73-s + 0.952·83-s + 1.99·97-s − 2.79·107-s + 0.376·109-s + 0.921·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.59·143-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.50·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2^2$ | \( 1 + 182 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 181 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 19 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 974 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 13715 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 5578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50834 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 109 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2126 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 59686 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 468 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 282202 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 145 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 503243 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 576 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 809 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 738971 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1336930 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 955 T + p^{3} T^{2} )^{2} \) |
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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
−10.58721981522263062475385356775, −9.967529610882726672002173208933, −9.885283206567734591998354297483, −9.175047023118522073651166482637, −8.501285543200427280445949743175, −8.198907832245753092237318240615, −7.79265067325963988854703261332, −7.48423981531874492966383470667, −6.71930909879560966546760227319, −6.07664594394550043970351128012, −5.84606913276363451092848651077, −5.24322196261803074117468901190, −4.68835443620870207294444637535, −4.08385856649487440722400659174, −3.40348196927632763780848398984, −2.89554620862120080874789329128, −2.02606227168322254483726754724, −1.57372916744254397069396739458, 0, 0,
1.57372916744254397069396739458, 2.02606227168322254483726754724, 2.89554620862120080874789329128, 3.40348196927632763780848398984, 4.08385856649487440722400659174, 4.68835443620870207294444637535, 5.24322196261803074117468901190, 5.84606913276363451092848651077, 6.07664594394550043970351128012, 6.71930909879560966546760227319, 7.48423981531874492966383470667, 7.79265067325963988854703261332, 8.198907832245753092237318240615, 8.501285543200427280445949743175, 9.175047023118522073651166482637, 9.885283206567734591998354297483, 9.967529610882726672002173208933, 10.58721981522263062475385356775
