| L(s) = 1 | − 2·5-s + 7-s − 2·11-s + 4·13-s − 3·17-s − 4·19-s − 2·23-s + 3·25-s − 29-s − 3·31-s − 2·35-s + 9·37-s − 7·41-s − 8·47-s − 9·49-s − 11·53-s + 4·55-s − 3·59-s + 10·61-s − 8·65-s + 5·67-s − 5·71-s + 8·73-s − 2·77-s − 14·79-s − 13·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.603·11-s + 1.10·13-s − 0.727·17-s − 0.917·19-s − 0.417·23-s + 3/5·25-s − 0.185·29-s − 0.538·31-s − 0.338·35-s + 1.47·37-s − 1.09·41-s − 1.16·47-s − 9/7·49-s − 1.51·53-s + 0.539·55-s − 0.390·59-s + 1.28·61-s − 0.992·65-s + 0.610·67-s − 0.593·71-s + 0.936·73-s − 0.227·77-s − 1.57·79-s − 1.42·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 132 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 116 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 102 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 144 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 170 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.216887867546573684669524334335, −8.125111924440308208641931118910, −7.49654204035658279344831072518, −7.26794457891808745717882071468, −6.66156772169152725959833876802, −6.47250147297528529127444795157, −6.12097262139449166300047833257, −5.68630774578045654389325157002, −5.03210572459989981311220633843, −4.95639415509300466858036466286, −4.36857115648337933624049307859, −4.09108229285627994637017246811, −3.56941221185371986249454273541, −3.37806509043737856736877147841, −2.50725038985008108395192898733, −2.46235175670899106774827910653, −1.44140646247733607545301804905, −1.36475244999116297851070167764, 0, 0,
1.36475244999116297851070167764, 1.44140646247733607545301804905, 2.46235175670899106774827910653, 2.50725038985008108395192898733, 3.37806509043737856736877147841, 3.56941221185371986249454273541, 4.09108229285627994637017246811, 4.36857115648337933624049307859, 4.95639415509300466858036466286, 5.03210572459989981311220633843, 5.68630774578045654389325157002, 6.12097262139449166300047833257, 6.47250147297528529127444795157, 6.66156772169152725959833876802, 7.26794457891808745717882071468, 7.49654204035658279344831072518, 8.125111924440308208641931118910, 8.216887867546573684669524334335
