| L(s) = 1 | − 2·5-s + 2·7-s + 8·11-s + 3·13-s − 2·17-s + 8·19-s − 2·23-s + 3·25-s + 13·29-s + 7·31-s − 4·35-s − 6·37-s + 3·41-s − 10·43-s − 5·47-s + 6·49-s − 8·53-s − 16·55-s − 4·59-s + 14·61-s − 6·65-s − 4·67-s + 5·71-s + 29·73-s + 16·77-s − 4·79-s − 24·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 2.41·11-s + 0.832·13-s − 0.485·17-s + 1.83·19-s − 0.417·23-s + 3/5·25-s + 2.41·29-s + 1.25·31-s − 0.676·35-s − 0.986·37-s + 0.468·41-s − 1.52·43-s − 0.729·47-s + 6/7·49-s − 1.09·53-s − 2.15·55-s − 0.520·59-s + 1.79·61-s − 0.744·65-s − 0.488·67-s + 0.593·71-s + 3.39·73-s + 1.82·77-s − 0.450·79-s − 2.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.915271917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.915271917\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 29 T + 352 T^{2} - 29 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.923769643146157676178375968490, −7.84707399761984396157813710519, −7.14489581671411361885464710153, −6.90898529115688709566366017144, −6.62634641992374752768862936009, −6.39337597931529919222221143614, −6.10199287219393165201701679281, −5.42761181959295018180435065252, −5.11083247686522509221237981106, −4.84551047265947340719162026368, −4.24755609493233515197054795986, −4.20855897884656671245321242527, −3.68936845469457780060311823362, −3.43834793725010218979562495934, −2.92906737971132218908179495127, −2.60999853986396129144298821915, −1.64704929413425993711186478908, −1.53459503631254466209190248204, −0.991002153158277674679225593400, −0.63811701482272834039898789466,
0.63811701482272834039898789466, 0.991002153158277674679225593400, 1.53459503631254466209190248204, 1.64704929413425993711186478908, 2.60999853986396129144298821915, 2.92906737971132218908179495127, 3.43834793725010218979562495934, 3.68936845469457780060311823362, 4.20855897884656671245321242527, 4.24755609493233515197054795986, 4.84551047265947340719162026368, 5.11083247686522509221237981106, 5.42761181959295018180435065252, 6.10199287219393165201701679281, 6.39337597931529919222221143614, 6.62634641992374752768862936009, 6.90898529115688709566366017144, 7.14489581671411361885464710153, 7.84707399761984396157813710519, 7.923769643146157676178375968490
