| L(s) = 1 | + 6·13-s + 4·17-s − 25-s − 20·29-s + 14·49-s + 4·53-s + 4·61-s + 32·79-s + 20·101-s + 8·103-s + 24·107-s − 12·113-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.66·13-s + 0.970·17-s − 1/5·25-s − 3.71·29-s + 2·49-s + 0.549·53-s + 0.512·61-s + 3.60·79-s + 1.99·101-s + 0.788·103-s + 2.32·107-s − 1.12·113-s + 1.63·121-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 + 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.621586022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.621586022\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.988879425210910685926001271408, −8.984052263852652518141509901286, −8.485634700026566492966918659682, −7.931310907318118276801025895921, −7.53211005729386736666643986721, −7.51208158328269920446402820038, −6.89576294912180826047189842108, −6.39353560416406445332392797216, −5.86795807085910800439538011407, −5.84329466848302182310044129887, −5.28379139250001909417545114428, −4.97401527434796915602183600662, −4.14872093646457107793043248390, −3.79445396745972790716454394092, −3.58514045685297104961909058507, −3.17828812749585374818105237016, −2.16358647606276954157858428996, −2.03900662860795572771315086828, −1.22069995112565507961852543400, −0.60457403218713909481485853069,
0.60457403218713909481485853069, 1.22069995112565507961852543400, 2.03900662860795572771315086828, 2.16358647606276954157858428996, 3.17828812749585374818105237016, 3.58514045685297104961909058507, 3.79445396745972790716454394092, 4.14872093646457107793043248390, 4.97401527434796915602183600662, 5.28379139250001909417545114428, 5.84329466848302182310044129887, 5.86795807085910800439538011407, 6.39353560416406445332392797216, 6.89576294912180826047189842108, 7.51208158328269920446402820038, 7.53211005729386736666643986721, 7.931310907318118276801025895921, 8.485634700026566492966918659682, 8.984052263852652518141509901286, 8.988879425210910685926001271408
