| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s + 4·13-s − 4·14-s + 5·16-s − 6·18-s + 4·19-s + 4·21-s − 4·22-s − 8·24-s + 2·25-s − 8·26-s + 4·27-s + 6·28-s − 12·29-s + 4·31-s − 6·32-s + 4·33-s + 9·36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 1.41·18-s + 0.917·19-s + 0.872·21-s − 0.852·22-s − 1.63·24-s + 2/5·25-s − 1.56·26-s + 0.769·27-s + 1.13·28-s − 2.22·29-s + 0.718·31-s − 1.06·32-s + 0.696·33-s + 3/2·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.762406373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.762406373\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.17998808305582310956809182042, −10.74433393642895475367766421449, −10.07863628872376377350248640291, −10.00110105167259080413415084245, −9.214714422376758870179865567047, −9.032638248930660592329284262401, −8.617718601682383826649694374982, −8.317512233996301506091531949146, −7.58015131040833785076560168365, −7.53358436179697876158945181235, −6.93845055513310391012329955760, −6.39419680149113395802921236634, −5.67761341162773234140283236806, −5.28117544514644935282791954773, −4.11932092109000370232963063510, −3.90942912380034135356675869307, −3.07281231850540375016606100273, −2.46076100486846570220703570998, −1.62497531587841803918805722140, −1.12602938014071185532476208364,
1.12602938014071185532476208364, 1.62497531587841803918805722140, 2.46076100486846570220703570998, 3.07281231850540375016606100273, 3.90942912380034135356675869307, 4.11932092109000370232963063510, 5.28117544514644935282791954773, 5.67761341162773234140283236806, 6.39419680149113395802921236634, 6.93845055513310391012329955760, 7.53358436179697876158945181235, 7.58015131040833785076560168365, 8.317512233996301506091531949146, 8.617718601682383826649694374982, 9.032638248930660592329284262401, 9.214714422376758870179865567047, 10.00110105167259080413415084245, 10.07863628872376377350248640291, 10.74433393642895475367766421449, 11.17998808305582310956809182042
