| L(s) = 1 | + 7-s + 9·13-s − 15·19-s − 10·25-s + 3·31-s + 11·37-s − 13·43-s − 6·49-s + 12·61-s − 5·67-s − 3·73-s − 17·79-s + 9·91-s − 24·97-s − 19·109-s + 22·121-s + 127-s + 131-s − 15·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 41·169-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 2.49·13-s − 3.44·19-s − 2·25-s + 0.538·31-s + 1.80·37-s − 1.98·43-s − 6/7·49-s + 1.53·61-s − 0.610·67-s − 0.351·73-s − 1.91·79-s + 0.943·91-s − 2.43·97-s − 1.81·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 1.30·133-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 + 3.15·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695486862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695486862\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.140531662285794971500046615681, −8.591136329343221754299480144843, −8.408645743331234944141388432961, −8.101643497405402838011373242662, −8.084623541462540238303941578809, −7.23374981002689541370255697442, −6.71330736461364241783891291740, −6.41894647877747878393988748706, −6.18706164906192565972816374562, −5.74894934170125048554235184319, −5.46854253744808558117497816264, −4.54282694794454783308357518015, −4.35629924131211705834691894046, −3.99835713751433763451568869440, −3.65301764510918033427104555916, −2.96732789724893050138109219792, −2.37998353963323020581031477000, −1.69743194326007219384634241561, −1.55344476624675286745246121697, −0.44082767595949334476307539968,
0.44082767595949334476307539968, 1.55344476624675286745246121697, 1.69743194326007219384634241561, 2.37998353963323020581031477000, 2.96732789724893050138109219792, 3.65301764510918033427104555916, 3.99835713751433763451568869440, 4.35629924131211705834691894046, 4.54282694794454783308357518015, 5.46854253744808558117497816264, 5.74894934170125048554235184319, 6.18706164906192565972816374562, 6.41894647877747878393988748706, 6.71330736461364241783891291740, 7.23374981002689541370255697442, 8.084623541462540238303941578809, 8.101643497405402838011373242662, 8.408645743331234944141388432961, 8.591136329343221754299480144843, 9.140531662285794971500046615681
