| L(s) = 1 | + 2·5-s + 2·9-s − 6·13-s − 25-s + 12·29-s − 12·37-s + 4·45-s − 16·47-s − 14·49-s + 12·61-s − 12·65-s + 24·67-s + 12·73-s − 5·81-s − 8·83-s + 12·97-s + 12·101-s − 12·117-s + 18·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s − 1.66·13-s − 1/5·25-s + 2.22·29-s − 1.97·37-s + 0.596·45-s − 2.33·47-s − 2·49-s + 1.53·61-s − 1.48·65-s + 2.93·67-s + 1.40·73-s − 5/9·81-s − 0.878·83-s + 1.21·97-s + 1.19·101-s − 1.10·117-s + 1.63·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.144180845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.144180845\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04550523434964221350431791893, −9.792937377881083293575322033069, −9.628517546118631800794722278491, −8.859463484345468218341805707242, −8.512001745118887951581839009846, −8.024899303462589330315294464003, −7.75558050714700163873008419714, −6.87765604520897280122427321441, −6.86546055063972996429447570295, −6.52208621393245152975290275549, −5.87885084718917339429083425466, −5.10774852424646912255425059314, −5.08857877220442595065199326266, −4.65102999398781231804203433239, −3.94275227180622519613045851516, −3.26951814974367224974595473256, −2.82677117847101712530097713933, −1.94897118320689200813379395203, −1.84779343068204927971982666045, −0.65121989537002453233270200583,
0.65121989537002453233270200583, 1.84779343068204927971982666045, 1.94897118320689200813379395203, 2.82677117847101712530097713933, 3.26951814974367224974595473256, 3.94275227180622519613045851516, 4.65102999398781231804203433239, 5.08857877220442595065199326266, 5.10774852424646912255425059314, 5.87885084718917339429083425466, 6.52208621393245152975290275549, 6.86546055063972996429447570295, 6.87765604520897280122427321441, 7.75558050714700163873008419714, 8.024899303462589330315294464003, 8.512001745118887951581839009846, 8.859463484345468218341805707242, 9.628517546118631800794722278491, 9.792937377881083293575322033069, 10.04550523434964221350431791893
