| L(s) = 1 | − 3-s + 5-s − 2·9-s − 15-s − 4·23-s − 4·25-s + 5·27-s − 7·31-s + 10·37-s − 2·45-s + 18·47-s + 12·49-s − 15·53-s + 5·67-s + 4·69-s − 5·71-s + 4·75-s + 81-s + 15·89-s + 7·93-s − 30·97-s − 15·103-s − 10·111-s − 12·113-s − 4·115-s − 11·121-s − 9·125-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.258·15-s − 0.834·23-s − 4/5·25-s + 0.962·27-s − 1.25·31-s + 1.64·37-s − 0.298·45-s + 2.62·47-s + 12/7·49-s − 2.06·53-s + 0.610·67-s + 0.481·69-s − 0.593·71-s + 0.461·75-s + 1/9·81-s + 1.58·89-s + 0.725·93-s − 3.04·97-s − 1.47·103-s − 0.949·111-s − 1.12·113-s − 0.373·115-s − 121-s − 0.804·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 107 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.77437978063122769066444294599, −7.15780150157986107438463036648, −6.65063802201351545623105104992, −6.25132400009762529877356440417, −5.79528594389715155791900523898, −5.53711899683432609886647353812, −5.25294405836126986308670599641, −4.39344230417460429496071175737, −4.14383348941596357231800666820, −3.61489976844520570484413755577, −2.79973139648444749287814196336, −2.46960851693567716447792188094, −1.80071691785605624859468146983, −0.959573350637239645440018495675, 0,
0.959573350637239645440018495675, 1.80071691785605624859468146983, 2.46960851693567716447792188094, 2.79973139648444749287814196336, 3.61489976844520570484413755577, 4.14383348941596357231800666820, 4.39344230417460429496071175737, 5.25294405836126986308670599641, 5.53711899683432609886647353812, 5.79528594389715155791900523898, 6.25132400009762529877356440417, 6.65063802201351545623105104992, 7.15780150157986107438463036648, 7.77437978063122769066444294599
