| L(s) = 1 | − 4·4-s + 36·11-s + 16·16-s − 46·19-s − 312·29-s − 170·31-s + 492·41-s − 144·44-s + 622·49-s − 1.58e3·59-s − 1.81e3·61-s − 64·64-s − 540·71-s + 184·76-s + 2.24e3·79-s + 396·89-s − 3.38e3·101-s − 1.18e3·109-s + 1.24e3·116-s − 1.69e3·121-s + 680·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.986·11-s + 1/4·16-s − 0.555·19-s − 1.99·29-s − 0.984·31-s + 1.87·41-s − 0.493·44-s + 1.81·49-s − 3.49·59-s − 3.80·61-s − 1/8·64-s − 0.902·71-s + 0.277·76-s + 3.19·79-s + 0.471·89-s − 3.33·101-s − 1.04·109-s + 0.998·116-s − 1.26·121-s + 0.492·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4330 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9601 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 23 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20365 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 156 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 85 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 p T + p^{3} T^{2} )( 1 + 12 p T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 122914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 124702 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 266425 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 907 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 497842 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 713518 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1123 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 549133 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 404482 T^{2} + p^{6} T^{4} \) |
| 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
−9.055455832060237230404706592805, −9.051030846765812384461340653480, −8.052334333130376691684436604482, −7.88042298201006568000471323522, −7.41291583584188488755137938927, −7.17971365768151308770570603195, −6.36773384814830232011755188541, −6.12915323667719069860385701574, −5.85453264262044858608782795585, −5.18478995185512754107150884997, −4.80965449919218334224360813860, −4.15541631334822718937275134324, −3.96820204405594301717472522322, −3.49382543185591561441561098237, −2.81121604093329720658235333156, −2.26425049342068580096804271970, −1.46750653432545916045186829165, −1.24379373231064953816497059673, 0, 0,
1.24379373231064953816497059673, 1.46750653432545916045186829165, 2.26425049342068580096804271970, 2.81121604093329720658235333156, 3.49382543185591561441561098237, 3.96820204405594301717472522322, 4.15541631334822718937275134324, 4.80965449919218334224360813860, 5.18478995185512754107150884997, 5.85453264262044858608782795585, 6.12915323667719069860385701574, 6.36773384814830232011755188541, 7.17971365768151308770570603195, 7.41291583584188488755137938927, 7.88042298201006568000471323522, 8.052334333130376691684436604482, 9.051030846765812384461340653480, 9.055455832060237230404706592805
