| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s − 26·9-s − 30·11-s + 80·16-s + 104·18-s + 120·22-s + 114·23-s − 142·29-s − 192·32-s − 312·36-s − 194·37-s + 198·43-s − 360·44-s − 456·46-s + 556·53-s + 568·58-s + 448·64-s + 170·67-s + 82·71-s + 832·72-s + 776·74-s − 1.57e3·79-s − 53·81-s − 792·86-s + 960·88-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.962·9-s − 0.822·11-s + 5/4·16-s + 1.36·18-s + 1.16·22-s + 1.03·23-s − 0.909·29-s − 1.06·32-s − 1.44·36-s − 0.861·37-s + 0.702·43-s − 1.23·44-s − 1.46·46-s + 1.44·53-s + 1.28·58-s + 7/8·64-s + 0.309·67-s + 0.137·71-s + 1.36·72-s + 1.21·74-s − 2.23·79-s − 0.0727·81-s − 0.993·86-s + 1.16·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 26 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 4142 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8818 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6550 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 57 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 71 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 51490 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 97 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 99510 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 99 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 100014 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 278 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 400650 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 406894 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 85 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 41 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 221366 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 785 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 324774 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 345238 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1791046 T^{2} + p^{6} T^{4} \) |
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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
−8.323325360389473426794646684045, −8.267987763154064434051073522152, −7.64922432143424339734720073356, −7.36499117443381620232016342382, −7.00063115386884039148941822942, −6.72502575243580093027859580502, −5.95755903112386144493539276396, −5.88550148858220614921347967934, −5.28994893341799630615080244950, −5.13422380598460319806046914275, −4.34713660648580608933577331503, −3.87468613202408957844555550417, −3.09999962910529292730342567656, −3.07059867827865191829972408283, −2.32349996811742551456805120328, −2.11122628608996326080267003496, −1.29483119312828002854551062814, −0.873425513988437019582098493004, 0, 0,
0.873425513988437019582098493004, 1.29483119312828002854551062814, 2.11122628608996326080267003496, 2.32349996811742551456805120328, 3.07059867827865191829972408283, 3.09999962910529292730342567656, 3.87468613202408957844555550417, 4.34713660648580608933577331503, 5.13422380598460319806046914275, 5.28994893341799630615080244950, 5.88550148858220614921347967934, 5.95755903112386144493539276396, 6.72502575243580093027859580502, 7.00063115386884039148941822942, 7.36499117443381620232016342382, 7.64922432143424339734720073356, 8.267987763154064434051073522152, 8.323325360389473426794646684045
