L(s) = 1 | + 2·5-s − 14·17-s − 7·25-s − 5·49-s + 30·61-s + 22·73-s − 28·85-s + 20·101-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 3.39·17-s − 7/5·25-s − 5/7·49-s + 3.84·61-s + 2.57·73-s − 3.03·85-s + 1.99·101-s + 3/11·121-s − 2.32·125-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.342901016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.342901016\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.253722077343463584061172115274, −8.789203814468410657458081175564, −8.242811409315443638822099337603, −8.020252288282131077617668391171, −7.48234465963839701103035147954, −6.83371142556904513206406479624, −6.82688651803480569959002336625, −6.32631396517094626619036615991, −6.08535659431749085987035216836, −5.57329956825449175714035398194, −4.95948616662942962246825509142, −4.91508712412840842965599357449, −4.24759561997224327573187415089, −3.78582158723682244956314339667, −3.59742293432366150954159283400, −2.50702635510614936132097004729, −2.25068134317812301064841386519, −2.15851289610000563168969067959, −1.33622192729485105473819296935, −0.36173015810441053873242161007,
0.36173015810441053873242161007, 1.33622192729485105473819296935, 2.15851289610000563168969067959, 2.25068134317812301064841386519, 2.50702635510614936132097004729, 3.59742293432366150954159283400, 3.78582158723682244956314339667, 4.24759561997224327573187415089, 4.91508712412840842965599357449, 4.95948616662942962246825509142, 5.57329956825449175714035398194, 6.08535659431749085987035216836, 6.32631396517094626619036615991, 6.82688651803480569959002336625, 6.83371142556904513206406479624, 7.48234465963839701103035147954, 8.020252288282131077617668391171, 8.242811409315443638822099337603, 8.789203814468410657458081175564, 9.253722077343463584061172115274