| L(s) = 1 | + 3-s + 5·7-s − 2·9-s + 2·11-s − 10·13-s − 3·17-s + 2·19-s + 5·21-s − 11·23-s − 2·27-s − 9·29-s − 6·31-s + 2·33-s − 12·37-s − 10·39-s − 4·41-s − 6·47-s + 8·49-s − 3·51-s − 53-s + 2·57-s + 14·59-s − 5·61-s − 10·63-s − 8·67-s − 11·69-s − 2·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s − 2/3·9-s + 0.603·11-s − 2.77·13-s − 0.727·17-s + 0.458·19-s + 1.09·21-s − 2.29·23-s − 0.384·27-s − 1.67·29-s − 1.07·31-s + 0.348·33-s − 1.97·37-s − 1.60·39-s − 0.624·41-s − 0.875·47-s + 8/7·49-s − 0.420·51-s − 0.137·53-s + 0.264·57-s + 1.82·59-s − 0.640·61-s − 1.25·63-s − 0.977·67-s − 1.32·69-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 11 T + 73 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 115 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 159 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 187 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 373 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
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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
−8.116614201032187966336244401496, −7.88528423778555297153566204033, −7.41749093613050304279654901068, −7.36212152040102579591131663691, −6.82891271655817307785250636237, −6.49425665666664869863378786812, −5.77583894084159838087199566627, −5.47695830624652788598464654383, −5.21776044684225107525648441632, −4.82843472176384337727465733967, −4.47763084185826970175800777007, −4.09176558341293282704534120405, −3.42509753135381068513173534223, −3.33393583863072467346875013831, −2.25556253433907257153572069024, −2.24157886352489509303527384583, −1.94306775672127537228610073543, −1.39392089734734951294124257392, 0, 0,
1.39392089734734951294124257392, 1.94306775672127537228610073543, 2.24157886352489509303527384583, 2.25556253433907257153572069024, 3.33393583863072467346875013831, 3.42509753135381068513173534223, 4.09176558341293282704534120405, 4.47763084185826970175800777007, 4.82843472176384337727465733967, 5.21776044684225107525648441632, 5.47695830624652788598464654383, 5.77583894084159838087199566627, 6.49425665666664869863378786812, 6.82891271655817307785250636237, 7.36212152040102579591131663691, 7.41749093613050304279654901068, 7.88528423778555297153566204033, 8.116614201032187966336244401496
