| L(s) = 1 | − 2·7-s − 4·13-s − 4·17-s − 8·19-s − 4·23-s + 4·29-s − 8·31-s − 4·37-s − 12·41-s + 8·43-s + 8·47-s + 3·49-s + 12·53-s + 8·59-s + 12·61-s − 16·67-s + 12·71-s + 4·73-s − 4·79-s − 9·81-s − 20·89-s + 8·91-s − 12·97-s + 12·101-s + 16·103-s − 16·107-s + 4·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.10·13-s − 0.970·17-s − 1.83·19-s − 0.834·23-s + 0.742·29-s − 1.43·31-s − 0.657·37-s − 1.87·41-s + 1.21·43-s + 1.16·47-s + 3/7·49-s + 1.64·53-s + 1.04·59-s + 1.53·61-s − 1.95·67-s + 1.42·71-s + 0.468·73-s − 0.450·79-s − 81-s − 2.11·89-s + 0.838·91-s − 1.21·97-s + 1.19·101-s + 1.57·103-s − 1.54·107-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.615231778091495348346849328066, −8.346135390691261229146037556740, −7.957917559360360332292014055983, −7.17163881570688473865439000383, −7.09469205153947435410375484788, −6.92810538526476435072768523651, −6.21144180293594010690559585904, −6.12007128634609921791500494127, −5.39706195677561815522561806423, −5.27693023530872518385519657807, −4.60093446893466198134098845149, −4.20105028102060935871931099403, −3.85199279054058084326272293738, −3.52280060844979710452524170987, −2.54313912566700600959037110610, −2.48647846848973479442346877592, −2.05148822211094492754286695184, −1.21552053391431321987771072747, 0, 0,
1.21552053391431321987771072747, 2.05148822211094492754286695184, 2.48647846848973479442346877592, 2.54313912566700600959037110610, 3.52280060844979710452524170987, 3.85199279054058084326272293738, 4.20105028102060935871931099403, 4.60093446893466198134098845149, 5.27693023530872518385519657807, 5.39706195677561815522561806423, 6.12007128634609921791500494127, 6.21144180293594010690559585904, 6.92810538526476435072768523651, 7.09469205153947435410375484788, 7.17163881570688473865439000383, 7.957917559360360332292014055983, 8.346135390691261229146037556740, 8.615231778091495348346849328066
