| L(s) = 1 | + 2·2-s + 3·4-s + 7-s + 4·8-s + 9-s + 2·14-s + 5·16-s + 2·18-s + 25-s + 3·28-s − 12·29-s + 6·32-s + 3·36-s + 4·37-s + 16·43-s + 49-s + 2·50-s + 12·53-s + 4·56-s − 24·58-s + 63-s + 7·64-s + 16·67-s + 4·72-s + 8·74-s − 32·79-s + 81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s + 1/3·9-s + 0.534·14-s + 5/4·16-s + 0.471·18-s + 1/5·25-s + 0.566·28-s − 2.22·29-s + 1.06·32-s + 1/2·36-s + 0.657·37-s + 2.43·43-s + 1/7·49-s + 0.282·50-s + 1.64·53-s + 0.534·56-s − 3.15·58-s + 0.125·63-s + 7/8·64-s + 1.95·67-s + 0.471·72-s + 0.929·74-s − 3.60·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.947123017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.947123017\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.731498057007750352966650002167, −8.319627980737908137853866947983, −7.51387576011083292544985077305, −7.27965583167809929942184453108, −7.10964669364204788106378776803, −6.14620005069006563789988203374, −5.78759423294267265835869539718, −5.60234946102839433104390934262, −4.64849576847169223614699526425, −4.57559453297699244994264276723, −3.71822253537069739639390686069, −3.54557244775390244119212987579, −2.47224864249386374740461957491, −2.16105217917237072156284139202, −1.12161915937701051533126457273,
1.12161915937701051533126457273, 2.16105217917237072156284139202, 2.47224864249386374740461957491, 3.54557244775390244119212987579, 3.71822253537069739639390686069, 4.57559453297699244994264276723, 4.64849576847169223614699526425, 5.60234946102839433104390934262, 5.78759423294267265835869539718, 6.14620005069006563789988203374, 7.10964669364204788106378776803, 7.27965583167809929942184453108, 7.51387576011083292544985077305, 8.319627980737908137853866947983, 8.731498057007750352966650002167
