| L(s) = 1 | + 2·5-s − 2·7-s − 7·11-s + 3·13-s − 5·17-s + 2·19-s − 2·23-s + 3·25-s + 3·29-s − 16·31-s − 4·35-s − 4·37-s − 2·41-s − 6·43-s − 3·47-s + 3·49-s − 10·53-s − 14·55-s − 16·59-s + 6·61-s + 6·65-s − 8·67-s − 16·71-s − 12·73-s + 14·77-s + 13·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 2.11·11-s + 0.832·13-s − 1.21·17-s + 0.458·19-s − 0.417·23-s + 3/5·25-s + 0.557·29-s − 2.87·31-s − 0.676·35-s − 0.657·37-s − 0.312·41-s − 0.914·43-s − 0.437·47-s + 3/7·49-s − 1.37·53-s − 1.88·55-s − 2.08·59-s + 0.768·61-s + 0.744·65-s − 0.977·67-s − 1.89·71-s − 1.40·73-s + 1.59·77-s + 1.46·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.639023573193590897685468693556, −8.544040435660303980020239051680, −7.86542886082277869450333294831, −7.64771707865377070173323032058, −7.06479948694465976360463360323, −6.93028845475709415349401719196, −6.17752843507424414426677055131, −6.12994451314874211855368659135, −5.49336908063962561400535550776, −5.41452933843588834477227168101, −4.70345998389568846415761000361, −4.57906171367077215620952219998, −3.61140589103804477795759636497, −3.47301113063531936381172497440, −2.72645083137326916993595109170, −2.62086274091989439536062087091, −1.65755016795417686027491303856, −1.62053753749064945525796922473, 0, 0,
1.62053753749064945525796922473, 1.65755016795417686027491303856, 2.62086274091989439536062087091, 2.72645083137326916993595109170, 3.47301113063531936381172497440, 3.61140589103804477795759636497, 4.57906171367077215620952219998, 4.70345998389568846415761000361, 5.41452933843588834477227168101, 5.49336908063962561400535550776, 6.12994451314874211855368659135, 6.17752843507424414426677055131, 6.93028845475709415349401719196, 7.06479948694465976360463360323, 7.64771707865377070173323032058, 7.86542886082277869450333294831, 8.544040435660303980020239051680, 8.639023573193590897685468693556
