| L(s) = 1 | + 2·5-s − 3·9-s + 12·17-s − 25-s − 6·45-s + 2·49-s + 4·53-s − 4·61-s + 9·81-s + 24·85-s − 36·109-s − 20·113-s − 6·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 36·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 2.91·17-s − 1/5·25-s − 0.894·45-s + 2/7·49-s + 0.549·53-s − 0.512·61-s + 81-s + 2.60·85-s − 3.44·109-s − 1.88·113-s − 0.545·121-s − 1.07·125-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 − 2.91·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412002795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412002795\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.40291222355631298305823974399, −10.08353888799581483933636788005, −9.464540754087187980012047302524, −9.171471501276194531373376297341, −8.328750961994828348739813899379, −7.929542490735932468142700417139, −7.43381007208035597056420893643, −6.57558114756947143484268138829, −5.97451966286629875811647616359, −5.44568706079483600893860377653, −5.22566784848492923050818654108, −3.99039936478397862704068269487, −3.24943376299631079642951352919, −2.55765980099239143499024710687, −1.33661798577425121421433395829,
1.33661798577425121421433395829, 2.55765980099239143499024710687, 3.24943376299631079642951352919, 3.99039936478397862704068269487, 5.22566784848492923050818654108, 5.44568706079483600893860377653, 5.97451966286629875811647616359, 6.57558114756947143484268138829, 7.43381007208035597056420893643, 7.929542490735932468142700417139, 8.328750961994828348739813899379, 9.171471501276194531373376297341, 9.464540754087187980012047302524, 10.08353888799581483933636788005, 10.40291222355631298305823974399
