| L(s) = 1 | − 3·3-s − 3·7-s + 3·9-s − 6·13-s + 17-s − 5·19-s + 9·21-s − 5·23-s + 7·25-s − 4·29-s + 5·31-s − 3·37-s + 18·39-s − 6·41-s + 43-s − 47-s − 49-s − 3·51-s + 15·57-s + 5·59-s − 3·61-s − 9·63-s + 13·67-s + 15·69-s + 3·71-s + 7·73-s − 21·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.13·7-s + 9-s − 1.66·13-s + 0.242·17-s − 1.14·19-s + 1.96·21-s − 1.04·23-s + 7/5·25-s − 0.742·29-s + 0.898·31-s − 0.493·37-s + 2.88·39-s − 0.937·41-s + 0.152·43-s − 0.145·47-s − 1/7·49-s − 0.420·51-s + 1.98·57-s + 0.650·59-s − 0.384·61-s − 1.13·63-s + 1.58·67-s + 1.80·69-s + 0.356·71-s + 0.819·73-s − 2.42·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 38 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 4 T - 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 190 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−17.2172607337, −16.9540137227, −16.5299061197, −16.0858127955, −15.5104850828, −14.8553947555, −14.4699669788, −13.7233385261, −12.9596376811, −12.6248343671, −12.0969853689, −11.8340611017, −11.0418334651, −10.6287640215, −9.93401785935, −9.70324939954, −8.78351586804, −8.00563162510, −7.11986820801, −6.52840870971, −6.19211669981, −5.26721021158, −4.89179602438, −3.76798168984, −2.51647120272, 0,
2.51647120272, 3.76798168984, 4.89179602438, 5.26721021158, 6.19211669981, 6.52840870971, 7.11986820801, 8.00563162510, 8.78351586804, 9.70324939954, 9.93401785935, 10.6287640215, 11.0418334651, 11.8340611017, 12.0969853689, 12.6248343671, 12.9596376811, 13.7233385261, 14.4699669788, 14.8553947555, 15.5104850828, 16.0858127955, 16.5299061197, 16.9540137227, 17.2172607337
