| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 2·9-s + 8·11-s + 5·16-s + 4·18-s + 16·22-s + 8·23-s − 2·25-s − 2·29-s + 6·32-s + 6·36-s + 16·37-s − 8·43-s + 24·44-s + 16·46-s − 4·50-s + 20·53-s − 4·58-s + 7·64-s + 4·67-s − 24·71-s + 8·72-s + 32·74-s − 12·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 2/3·9-s + 2.41·11-s + 5/4·16-s + 0.942·18-s + 3.41·22-s + 1.66·23-s − 2/5·25-s − 0.371·29-s + 1.06·32-s + 36-s + 2.63·37-s − 1.21·43-s + 3.61·44-s + 2.35·46-s − 0.565·50-s + 2.74·53-s − 0.525·58-s + 7/8·64-s + 0.488·67-s − 2.84·71-s + 0.942·72-s + 3.71·74-s − 1.35·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.05386238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.05386238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 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
−9.098249465208413752011131908983, −8.628218145104970601495367490310, −8.178223084381374482344902768316, −7.64992267808901979928473490986, −7.15342435605796983575784659068, −6.90949152981069454555699471435, −6.85526230974732777093571505747, −6.17226094453500522181639238914, −5.85693339879420418905739213551, −5.66000084538915797066648966925, −4.97131202208840324222402099455, −4.48469233465941009213470623592, −4.20539387683931015972668882499, −4.08957129102675588160954892117, −3.30753881235218701135331932831, −3.19943540270961914778283679601, −2.45195607530105577788056949440, −1.91886800239524716367499283462, −1.24151604199505148875982273453, −0.986952263311565642446113258526,
0.986952263311565642446113258526, 1.24151604199505148875982273453, 1.91886800239524716367499283462, 2.45195607530105577788056949440, 3.19943540270961914778283679601, 3.30753881235218701135331932831, 4.08957129102675588160954892117, 4.20539387683931015972668882499, 4.48469233465941009213470623592, 4.97131202208840324222402099455, 5.66000084538915797066648966925, 5.85693339879420418905739213551, 6.17226094453500522181639238914, 6.85526230974732777093571505747, 6.90949152981069454555699471435, 7.15342435605796983575784659068, 7.64992267808901979928473490986, 8.178223084381374482344902768316, 8.628218145104970601495367490310, 9.098249465208413752011131908983
