| L(s) = 1 | + 2·2-s + 2·3-s − 4-s + 4·6-s − 8·8-s + 9-s + 11-s − 2·12-s − 7·16-s + 8·17-s + 2·18-s + 2·22-s − 16·24-s − 6·25-s − 4·27-s − 12·29-s + 20·31-s + 14·32-s + 2·33-s + 16·34-s − 36-s − 12·37-s + 8·41-s − 44-s − 14·48-s + 49-s − 12·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 1/2·4-s + 1.63·6-s − 2.82·8-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 7/4·16-s + 1.94·17-s + 0.471·18-s + 0.426·22-s − 3.26·24-s − 6/5·25-s − 0.769·27-s − 2.22·29-s + 3.59·31-s + 2.47·32-s + 0.348·33-s + 2.74·34-s − 1/6·36-s − 1.97·37-s + 1.24·41-s − 0.150·44-s − 2.02·48-s + 1/7·49-s − 1.69·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.671852040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.671852040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.576763002125084546471923487790, −7.87739468442856243522484593458, −7.82867506007281568614416512666, −7.07007000345216400828661927108, −6.31251542982324707519501957045, −5.96443978110022033983848919969, −5.45217811977635791883277118514, −5.22894511431647767966156393918, −4.45772251821753336888632495633, −4.07930883034575373640745113861, −3.58922471493276128945836920884, −3.26888344602981049454952325554, −2.78349816157540378309057592549, −1.96167877918167104055842099400, −0.75006805308184274174858297412,
0.75006805308184274174858297412, 1.96167877918167104055842099400, 2.78349816157540378309057592549, 3.26888344602981049454952325554, 3.58922471493276128945836920884, 4.07930883034575373640745113861, 4.45772251821753336888632495633, 5.22894511431647767966156393918, 5.45217811977635791883277118514, 5.96443978110022033983848919969, 6.31251542982324707519501957045, 7.07007000345216400828661927108, 7.82867506007281568614416512666, 7.87739468442856243522484593458, 8.576763002125084546471923487790
