| L(s) = 1 | − 2·5-s + 2·13-s + 6·17-s − 25-s + 8·29-s + 14·37-s + 16·41-s + 18·53-s − 4·65-s + 22·73-s − 9·81-s − 12·85-s + 26·97-s − 12·109-s − 2·113-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.554·13-s + 1.45·17-s − 1/5·25-s + 1.48·29-s + 2.30·37-s + 2.49·41-s + 2.47·53-s − 0.496·65-s + 2.57·73-s − 81-s − 1.30·85-s + 2.63·97-s − 1.14·109-s − 0.188·113-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298856930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298856930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−9.825958827483972887153904020253, −9.568240642331605961562944213372, −8.868856216771511160194219694798, −8.800093670867458467264138813393, −7.955683167132138330383180203104, −7.938343535944210077954830740999, −7.62047579371549581888179827572, −7.12374153116788240812881369957, −6.43879918036070445624435376550, −6.26431745713657614381074924810, −5.58255744935582529467512340996, −5.42280719842764906931291849703, −4.55994791326550828512608548662, −4.31530874717417801948092227188, −3.77087942110642421376506493881, −3.41917159381795126682362195226, −2.63468761777757803659109401171, −2.35932758148223026856062009519, −1.07560883208194430273964421974, −0.820320506913134344902456481871,
0.820320506913134344902456481871, 1.07560883208194430273964421974, 2.35932758148223026856062009519, 2.63468761777757803659109401171, 3.41917159381795126682362195226, 3.77087942110642421376506493881, 4.31530874717417801948092227188, 4.55994791326550828512608548662, 5.42280719842764906931291849703, 5.58255744935582529467512340996, 6.26431745713657614381074924810, 6.43879918036070445624435376550, 7.12374153116788240812881369957, 7.62047579371549581888179827572, 7.938343535944210077954830740999, 7.955683167132138330383180203104, 8.800093670867458467264138813393, 8.868856216771511160194219694798, 9.568240642331605961562944213372, 9.825958827483972887153904020253
