| L(s) = 1 | − 2·2-s + 3·4-s − 6·5-s − 4·8-s + 12·10-s − 2·11-s + 5·16-s + 6·19-s − 18·20-s + 4·22-s + 2·23-s + 19·25-s + 8·29-s + 6·31-s − 6·32-s + 2·37-s − 12·38-s + 24·40-s − 18·41-s + 4·43-s − 6·44-s − 4·46-s − 12·47-s − 38·50-s + 16·53-s + 12·55-s − 16·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 2.68·5-s − 1.41·8-s + 3.79·10-s − 0.603·11-s + 5/4·16-s + 1.37·19-s − 4.02·20-s + 0.852·22-s + 0.417·23-s + 19/5·25-s + 1.48·29-s + 1.07·31-s − 1.06·32-s + 0.328·37-s − 1.94·38-s + 3.79·40-s − 2.81·41-s + 0.609·43-s − 0.904·44-s − 0.589·46-s − 1.75·47-s − 5.37·50-s + 2.19·53-s + 1.61·55-s − 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 39 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 18 T + 155 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 24 T + 258 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
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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
−8.407402061190923619451755323100, −8.286792238368949617331425222827, −7.962035694241662108664010302925, −7.70760115156622673825612550338, −7.34433610062321642573310067817, −7.03639357661927983510146439714, −6.53300645337732712080617026304, −6.36881981196400468140037820839, −5.50106927532037167917415566171, −5.12045544713022865077138295735, −4.60330205049140923825091341255, −4.35365403185281411574788853838, −3.60216949801363332690105970281, −3.32796979722954235934600701890, −2.90954569201735560829534542417, −2.54995284564172312716117646968, −1.35050129198020884482127952700, −1.12483444901762346494863767768, 0, 0,
1.12483444901762346494863767768, 1.35050129198020884482127952700, 2.54995284564172312716117646968, 2.90954569201735560829534542417, 3.32796979722954235934600701890, 3.60216949801363332690105970281, 4.35365403185281411574788853838, 4.60330205049140923825091341255, 5.12045544713022865077138295735, 5.50106927532037167917415566171, 6.36881981196400468140037820839, 6.53300645337732712080617026304, 7.03639357661927983510146439714, 7.34433610062321642573310067817, 7.70760115156622673825612550338, 7.962035694241662108664010302925, 8.286792238368949617331425222827, 8.407402061190923619451755323100
