| L(s) = 1 | − 6·7-s + 4·13-s + 2·19-s − 3·25-s − 20·31-s + 8·37-s − 2·43-s + 13·49-s − 14·61-s − 24·67-s − 6·73-s + 8·79-s − 24·91-s − 28·97-s − 12·103-s − 24·109-s − 15·121-s + 127-s + 131-s − 12·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 1.10·13-s + 0.458·19-s − 3/5·25-s − 3.59·31-s + 1.31·37-s − 0.304·43-s + 13/7·49-s − 1.79·61-s − 2.93·67-s − 0.702·73-s + 0.900·79-s − 2.51·91-s − 2.84·97-s − 1.18·103-s − 2.29·109-s − 1.36·121-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 87 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.786690907053672978320749134025, −8.337253088224482413108269863201, −7.61650110513974439422101562544, −7.60151955653187393737520991404, −7.12681127319610501626511092750, −6.62108100873236031683883377356, −6.33066898878993511913220552248, −6.03310004115110628402293225783, −5.58194770150736054319584499483, −5.40527365063644770050122165934, −4.64052041835781739077705813380, −3.98540664321105212145219076687, −3.82855859415052343694498541938, −3.39770293688043109133819286443, −2.92903486699149523204204510571, −2.63333420825581807648679559822, −1.67111828486453016703581803769, −1.32969891742440405173778931249, 0, 0,
1.32969891742440405173778931249, 1.67111828486453016703581803769, 2.63333420825581807648679559822, 2.92903486699149523204204510571, 3.39770293688043109133819286443, 3.82855859415052343694498541938, 3.98540664321105212145219076687, 4.64052041835781739077705813380, 5.40527365063644770050122165934, 5.58194770150736054319584499483, 6.03310004115110628402293225783, 6.33066898878993511913220552248, 6.62108100873236031683883377356, 7.12681127319610501626511092750, 7.60151955653187393737520991404, 7.61650110513974439422101562544, 8.337253088224482413108269863201, 8.786690907053672978320749134025
