| L(s) = 1 | + 2-s − 7-s − 13-s − 14-s + 17-s − 19-s + 23-s + 2·25-s − 26-s + 29-s − 32-s + 34-s − 38-s + 41-s + 46-s + 2·50-s + 58-s + 59-s − 61-s − 64-s − 79-s + 82-s + 83-s + 91-s − 97-s + 2·103-s − 4·107-s + ⋯ |
| L(s) = 1 | + 2-s − 7-s − 13-s − 14-s + 17-s − 19-s + 23-s + 2·25-s − 26-s + 29-s − 32-s + 34-s − 38-s + 41-s + 46-s + 2·50-s + 58-s + 59-s − 61-s − 64-s − 79-s + 82-s + 83-s + 91-s − 97-s + 2·103-s − 4·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 859329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.280651094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280651094\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 103 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 19 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 29 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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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
−10.77261549186022846070836068324, −10.04520232982095257651501639934, −9.560699256059565747450522245060, −9.453046778798123245451729413043, −8.708758436645538016550580608567, −8.555809977384639433432940183809, −7.932477664123689788445864580517, −7.32420320383045325708279180575, −6.94620883127531000143676040847, −6.74838133129326370058408346152, −6.06210862698941804644260382126, −5.68229934267069263904404091702, −5.07090324882585843755023687659, −4.74780865900708731254552737730, −4.39280252164025836403199604291, −3.79285988259939986749569744693, −3.02442426386372928453242059648, −2.97671953618363196033058754047, −2.13516838096298957613053787995, −1.01777354278783439579145493173,
1.01777354278783439579145493173, 2.13516838096298957613053787995, 2.97671953618363196033058754047, 3.02442426386372928453242059648, 3.79285988259939986749569744693, 4.39280252164025836403199604291, 4.74780865900708731254552737730, 5.07090324882585843755023687659, 5.68229934267069263904404091702, 6.06210862698941804644260382126, 6.74838133129326370058408346152, 6.94620883127531000143676040847, 7.32420320383045325708279180575, 7.932477664123689788445864580517, 8.555809977384639433432940183809, 8.708758436645538016550580608567, 9.453046778798123245451729413043, 9.560699256059565747450522245060, 10.04520232982095257651501639934, 10.77261549186022846070836068324
