| L(s) = 1 | − 3-s + 2·4-s − 5-s − 7-s − 11-s − 2·12-s − 13-s + 15-s + 3·16-s − 2·20-s + 21-s − 2·28-s + 33-s + 35-s + 39-s − 41-s − 43-s − 2·44-s − 3·48-s − 2·52-s + 4·53-s + 55-s − 59-s + 2·60-s − 61-s + 4·64-s + 65-s + ⋯ |
| L(s) = 1 | − 3-s + 2·4-s − 5-s − 7-s − 11-s − 2·12-s − 13-s + 15-s + 3·16-s − 2·20-s + 21-s − 2·28-s + 33-s + 35-s + 39-s − 41-s − 43-s − 2·44-s − 3·48-s − 2·52-s + 4·53-s + 55-s − 59-s + 2·60-s − 61-s + 4·64-s + 65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3263700811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3263700811\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 131 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$ | \( ( 1 - T )^{4} \) |
| 59 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−13.49115710230681409798432755353, −13.22375212491396362504805975671, −12.29259752204418281575577204325, −12.21274170830972099991167298192, −11.66281031150107468242863614059, −11.55149329870096178406183095752, −10.63275133986296812825530566576, −10.50955681500881916104758058698, −10.04012334184021374516029299482, −9.218449254119487021935505079960, −8.157385197801971307650196111733, −7.81691846039840170699782085388, −7.11096588269597252147591005558, −6.88405597292774965520044607618, −6.14622486610113318867757722039, −5.62217519851476845691626484153, −5.02577259089436039660102904084, −3.75376533239769561770804369848, −3.03544317976724495517564615348, −2.26288523295514045403412685904,
2.26288523295514045403412685904, 3.03544317976724495517564615348, 3.75376533239769561770804369848, 5.02577259089436039660102904084, 5.62217519851476845691626484153, 6.14622486610113318867757722039, 6.88405597292774965520044607618, 7.11096588269597252147591005558, 7.81691846039840170699782085388, 8.157385197801971307650196111733, 9.218449254119487021935505079960, 10.04012334184021374516029299482, 10.50955681500881916104758058698, 10.63275133986296812825530566576, 11.55149329870096178406183095752, 11.66281031150107468242863614059, 12.21274170830972099991167298192, 12.29259752204418281575577204325, 13.22375212491396362504805975671, 13.49115710230681409798432755353
