| L(s) = 1 | − 8·11-s − 24·19-s + 16·41-s − 24·49-s − 8·59-s − 18·81-s − 8·89-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 192·209-s + 211-s + ⋯ |
| L(s) = 1 | − 2.41·11-s − 5.50·19-s + 2.49·41-s − 3.42·49-s − 1.04·59-s − 2·81-s − 0.847·89-s − 0.363·121-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 + 0.923·169-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 + 13.2·209-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.07775067471484393274765844554, −5.82918661716317186335975071491, −5.72451224869772889238970180177, −5.42196718003033120349926344005, −5.31483315895409588978374581713, −4.94679511059305142987727395807, −4.91413888755313174254911160467, −4.64580091158980977231455924302, −4.45405156077165604980132944207, −4.39853159033005422818789055526, −4.06165613807949371063861018889, −4.05664473636801749738831776862, −3.94286355214270741054127983679, −3.40595804995048975797934896063, −3.15295046556227202226141619021, −3.13169536142709483103181628056, −2.85166243272533646525486940196, −2.50610155220190590941183348908, −2.37732851806187775243003707669, −2.23248042853116757195136593935, −2.15799997858690678273641235582, −1.70331265173492905390901899734, −1.67052997609667589276355101763, −1.03914068266068484152548731698, −1.02291908976022793339981388392, 0, 0, 0, 0,
1.02291908976022793339981388392, 1.03914068266068484152548731698, 1.67052997609667589276355101763, 1.70331265173492905390901899734, 2.15799997858690678273641235582, 2.23248042853116757195136593935, 2.37732851806187775243003707669, 2.50610155220190590941183348908, 2.85166243272533646525486940196, 3.13169536142709483103181628056, 3.15295046556227202226141619021, 3.40595804995048975797934896063, 3.94286355214270741054127983679, 4.05664473636801749738831776862, 4.06165613807949371063861018889, 4.39853159033005422818789055526, 4.45405156077165604980132944207, 4.64580091158980977231455924302, 4.91413888755313174254911160467, 4.94679511059305142987727395807, 5.31483315895409588978374581713, 5.42196718003033120349926344005, 5.72451224869772889238970180177, 5.82918661716317186335975071491, 6.07775067471484393274765844554
