| L(s) = 1 | − 2·3-s − 2·5-s + 2·11-s + 2·13-s + 4·15-s + 8·17-s − 2·19-s + 2·23-s + 3·25-s + 2·27-s + 12·29-s + 10·31-s − 4·33-s − 4·39-s + 8·41-s − 18·43-s + 16·47-s − 2·49-s − 16·51-s − 8·53-s − 4·55-s + 4·57-s − 6·59-s − 4·61-s − 4·65-s − 12·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.603·11-s + 0.554·13-s + 1.03·15-s + 1.94·17-s − 0.458·19-s + 0.417·23-s + 3/5·25-s + 0.384·27-s + 2.22·29-s + 1.79·31-s − 0.696·33-s − 0.640·39-s + 1.24·41-s − 2.74·43-s + 2.33·47-s − 2/7·49-s − 2.24·51-s − 1.09·53-s − 0.539·55-s + 0.529·57-s − 0.781·59-s − 0.512·61-s − 0.496·65-s − 1.46·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001045296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001045296\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 18 T + 164 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 260 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.97600575780112934316907505504, −10.90408677099640618740331508210, −10.20414253020792576095761356254, −10.02924581371377934703209868279, −9.368171480062089548096755888861, −8.793363143840068806807251551510, −8.274658006708178020450122064181, −8.060210896072692909797751119787, −7.53844637418609047564072990656, −6.83133973832531209909933199725, −6.34776039853551283174323004610, −6.23744849259013932365528684669, −5.37276639251637763627303023232, −5.16920999808620702659828679307, −4.34215244154987436195976911636, −4.09043065945765163757391542059, −3.04747144414217232239792058212, −2.95818306668519194005910648112, −1.38230213712586315015274173446, −0.72301461500310057444827065062,
0.72301461500310057444827065062, 1.38230213712586315015274173446, 2.95818306668519194005910648112, 3.04747144414217232239792058212, 4.09043065945765163757391542059, 4.34215244154987436195976911636, 5.16920999808620702659828679307, 5.37276639251637763627303023232, 6.23744849259013932365528684669, 6.34776039853551283174323004610, 6.83133973832531209909933199725, 7.53844637418609047564072990656, 8.060210896072692909797751119787, 8.274658006708178020450122064181, 8.793363143840068806807251551510, 9.368171480062089548096755888861, 10.02924581371377934703209868279, 10.20414253020792576095761356254, 10.90408677099640618740331508210, 10.97600575780112934316907505504
