L(s) = 1 | − 2·2-s + 3·4-s − 5-s − 4·8-s + 2·10-s + 5·16-s + 2·19-s − 3·20-s + 23-s − 3·31-s − 6·32-s − 4·38-s + 4·40-s − 2·46-s + 2·47-s + 49-s + 2·53-s + 3·61-s + 6·62-s + 7·64-s + 6·76-s − 3·79-s − 5·80-s + 3·83-s + 3·92-s − 4·94-s − 2·95-s + ⋯ |
L(s) = 1 | − 2·2-s + 3·4-s − 5-s − 4·8-s + 2·10-s + 5·16-s + 2·19-s − 3·20-s + 23-s − 3·31-s − 6·32-s − 4·38-s + 4·40-s − 2·46-s + 2·47-s + 49-s + 2·53-s + 3·61-s + 6·62-s + 7·64-s + 6·76-s − 3·79-s − 5·80-s + 3·83-s + 3·92-s − 4·94-s − 2·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4833150574\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4833150574\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.892129304618549294684432002935, −8.795877625995565746022054425171, −8.322253063691103526232715978680, −7.937226770852492521139951155570, −7.46180425083176276816000841178, −7.27912523533879543231904159018, −7.00888502137491555693564544509, −6.97886813211644625743582875593, −5.98611218435903779852507887347, −5.76722528973512968031969162656, −5.35751382373342910778605412372, −5.11847098845601104547859308480, −4.00060221469824650592778499589, −3.85773602517784727052389470813, −3.34875778136731301969430803148, −2.94601141062039916783181935240, −2.30070042155526006335616633926, −1.95272012553594927667455717285, −1.07774004749337072189208513184, −0.70540924740832342775693901353,
0.70540924740832342775693901353, 1.07774004749337072189208513184, 1.95272012553594927667455717285, 2.30070042155526006335616633926, 2.94601141062039916783181935240, 3.34875778136731301969430803148, 3.85773602517784727052389470813, 4.00060221469824650592778499589, 5.11847098845601104547859308480, 5.35751382373342910778605412372, 5.76722528973512968031969162656, 5.98611218435903779852507887347, 6.97886813211644625743582875593, 7.00888502137491555693564544509, 7.27912523533879543231904159018, 7.46180425083176276816000841178, 7.937226770852492521139951155570, 8.322253063691103526232715978680, 8.795877625995565746022054425171, 8.892129304618549294684432002935