| L(s) = 1 | − 3-s + 2·5-s − 7-s + 3·11-s − 7·13-s − 2·15-s − 3·17-s + 7·19-s + 21-s − 2·23-s + 3·25-s − 2·27-s − 6·29-s − 7·31-s − 3·33-s − 2·35-s + 8·37-s + 7·39-s − 9·41-s + 4·43-s + 18·47-s − 8·49-s + 3·51-s − 12·53-s + 6·55-s − 7·57-s − 18·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 0.904·11-s − 1.94·13-s − 0.516·15-s − 0.727·17-s + 1.60·19-s + 0.218·21-s − 0.417·23-s + 3/5·25-s − 0.384·27-s − 1.11·29-s − 1.25·31-s − 0.522·33-s − 0.338·35-s + 1.31·37-s + 1.12·39-s − 1.40·41-s + 0.609·43-s + 2.62·47-s − 8/7·49-s + 0.420·51-s − 1.64·53-s + 0.809·55-s − 0.927·57-s − 2.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 154 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−7.55291806169122076557142835551, −7.45848393910505131159186025650, −6.95546336171878306762377133017, −6.78942993953938706192516552539, −6.20276387456542630006952107812, −5.96315535482321810858216337589, −5.63402546314421436288590255920, −5.44183856446622665373114290302, −4.91510503580694977412364053779, −4.50656033201646925954566939121, −4.37275139481444825531417408737, −3.75127232953062035200974254214, −3.12727636159857635369696757897, −3.08096277719970831573947335406, −2.37068626441754443045267673256, −2.04645572360854975937822994843, −1.53138470400785060820374908749, −1.10791543709437474234977856114, 0, 0,
1.10791543709437474234977856114, 1.53138470400785060820374908749, 2.04645572360854975937822994843, 2.37068626441754443045267673256, 3.08096277719970831573947335406, 3.12727636159857635369696757897, 3.75127232953062035200974254214, 4.37275139481444825531417408737, 4.50656033201646925954566939121, 4.91510503580694977412364053779, 5.44183856446622665373114290302, 5.63402546314421436288590255920, 5.96315535482321810858216337589, 6.20276387456542630006952107812, 6.78942993953938706192516552539, 6.95546336171878306762377133017, 7.45848393910505131159186025650, 7.55291806169122076557142835551
