| L(s) = 1 | + 4-s + 2·7-s + 9-s + 16-s + 12·23-s + 2·25-s + 2·28-s + 36-s + 4·37-s + 4·43-s − 3·49-s + 2·63-s + 64-s − 8·67-s − 12·71-s − 20·79-s + 81-s + 12·92-s + 2·100-s + 24·107-s − 20·109-s + 2·112-s − 12·113-s + 121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 1/3·9-s + 1/4·16-s + 2.50·23-s + 2/5·25-s + 0.377·28-s + 1/6·36-s + 0.657·37-s + 0.609·43-s − 3/7·49-s + 0.251·63-s + 1/8·64-s − 0.977·67-s − 1.42·71-s − 2.25·79-s + 1/9·81-s + 1.25·92-s + 1/5·100-s + 2.32·107-s − 1.91·109-s + 0.188·112-s − 1.12·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.372238099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.372238099\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.961684234706022251677551424357, −8.684908850131452907045183858594, −8.040059960932505691174498165909, −7.50962167165330375562844156763, −7.23476794814776921954312749019, −6.71629543572186837926893016369, −6.20883336071643376251210250265, −5.54164136294502428930682083604, −5.08836479265341328600038231395, −4.54769742698713509305480401642, −4.05667893012187444424704505059, −3.03102667983335410829013472586, −2.81558505871621093504501002145, −1.74339243545524085960081520464, −1.07587563756711607113330517913,
1.07587563756711607113330517913, 1.74339243545524085960081520464, 2.81558505871621093504501002145, 3.03102667983335410829013472586, 4.05667893012187444424704505059, 4.54769742698713509305480401642, 5.08836479265341328600038231395, 5.54164136294502428930682083604, 6.20883336071643376251210250265, 6.71629543572186837926893016369, 7.23476794814776921954312749019, 7.50962167165330375562844156763, 8.040059960932505691174498165909, 8.684908850131452907045183858594, 8.961684234706022251677551424357
