| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s − 2·13-s − 6·17-s + 2·19-s − 2·23-s + 3·25-s + 6·31-s − 4·35-s + 2·37-s + 4·41-s + 4·43-s − 18·47-s + 6·49-s + 20·53-s − 8·55-s − 12·59-s + 4·61-s − 4·65-s + 4·67-s − 16·71-s + 8·73-s + 8·77-s + 18·79-s + 8·83-s − 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s − 0.417·23-s + 3/5·25-s + 1.07·31-s − 0.676·35-s + 0.328·37-s + 0.624·41-s + 0.609·43-s − 2.62·47-s + 6/7·49-s + 2.74·53-s − 1.07·55-s − 1.56·59-s + 0.512·61-s − 0.496·65-s + 0.488·67-s − 1.89·71-s + 0.936·73-s + 0.911·77-s + 2.02·79-s + 0.878·83-s − 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.315032085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.315032085\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.291425626133885131447471639911, −7.914743384743847789396823601899, −7.44310471144744002506651548979, −7.40347985127455587811974426786, −6.65475252304571903404839488759, −6.60149528994467914124537522835, −6.07034111250826551655594039371, −5.98531967112536079421426304418, −5.39579491858308401601151102235, −5.04003267608883251888328501931, −4.66668068557676041840456934077, −4.50243423219192801440347981329, −3.76717152926317884945505871777, −3.40486913657657723893497113046, −2.90889962204919948698199456394, −2.54607190849416688930504152867, −2.05164940416757287295694611362, −1.97337525054549237646673564871, −0.849199080692888993655681324053, −0.48090025765838999427154838063,
0.48090025765838999427154838063, 0.849199080692888993655681324053, 1.97337525054549237646673564871, 2.05164940416757287295694611362, 2.54607190849416688930504152867, 2.90889962204919948698199456394, 3.40486913657657723893497113046, 3.76717152926317884945505871777, 4.50243423219192801440347981329, 4.66668068557676041840456934077, 5.04003267608883251888328501931, 5.39579491858308401601151102235, 5.98531967112536079421426304418, 6.07034111250826551655594039371, 6.60149528994467914124537522835, 6.65475252304571903404839488759, 7.40347985127455587811974426786, 7.44310471144744002506651548979, 7.914743384743847789396823601899, 8.291425626133885131447471639911
