| L(s) = 1 | + 2-s − 8-s + 9-s + 4·13-s − 16-s + 17-s + 18-s − 25-s + 4·26-s + 34-s − 50-s − 2·53-s + 64-s − 72-s − 2·89-s − 2·101-s − 4·104-s − 2·106-s + 4·117-s + 121-s + 127-s + 128-s + 131-s − 136-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s + 9-s + 4·13-s − 16-s + 17-s + 18-s − 25-s + 4·26-s + 34-s − 50-s − 2·53-s + 64-s − 72-s − 2·89-s − 2·101-s − 4·104-s − 2·106-s + 4·117-s + 121-s + 127-s + 128-s + 131-s − 136-s + 137-s + 139-s − 144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.885624111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.885624111\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−8.870431060253199522193022083744, −8.612420346822928727030383580695, −8.176187904380130979774471702948, −8.052473101106199130887091473846, −7.56244060526558704183467388904, −6.78421467486257145275805207514, −6.76629422866384744025966600858, −6.17516738020270335602904850429, −5.95236142671556961796155853963, −5.62264660835330541231651592472, −5.36332984954896989788059762674, −4.47887448849527279390066982580, −4.43172519368160728415277307891, −3.77334980129548481202183683420, −3.72845319858894840005191816088, −3.24897868620786090643291514224, −2.87969008200034055028423249611, −1.87312827483466508110981069627, −1.43444400753966688670149187750, −1.02368798150210056960391418729,
1.02368798150210056960391418729, 1.43444400753966688670149187750, 1.87312827483466508110981069627, 2.87969008200034055028423249611, 3.24897868620786090643291514224, 3.72845319858894840005191816088, 3.77334980129548481202183683420, 4.43172519368160728415277307891, 4.47887448849527279390066982580, 5.36332984954896989788059762674, 5.62264660835330541231651592472, 5.95236142671556961796155853963, 6.17516738020270335602904850429, 6.76629422866384744025966600858, 6.78421467486257145275805207514, 7.56244060526558704183467388904, 8.052473101106199130887091473846, 8.176187904380130979774471702948, 8.612420346822928727030383580695, 8.870431060253199522193022083744
