| L(s) = 1 | − 7-s + 13-s + 5·19-s − 25-s − 6·31-s − 5·37-s + 2·43-s − 11·49-s + 61-s + 14·67-s + 19·73-s − 79-s − 91-s − 14·97-s + 17·103-s + 13·109-s + 5·121-s + 127-s + 131-s − 5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.277·13-s + 1.14·19-s − 1/5·25-s − 1.07·31-s − 0.821·37-s + 0.304·43-s − 1.57·49-s + 0.128·61-s + 1.71·67-s + 2.22·73-s − 0.112·79-s − 0.104·91-s − 1.42·97-s + 1.67·103-s + 1.24·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.433·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 642816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.679346813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679346813\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.278659456837051791880753430110, −8.009178615776485282232066036514, −7.41340325354197965108863025841, −7.04066013021810195506913088820, −6.61564020931240255898157210391, −6.11277041848327263557342668998, −5.56250138480305174001193584134, −5.21135737120813885125080962772, −4.70149803454162952737464687808, −3.99415195845520306957084764823, −3.43634660745995933925762791811, −3.18235696314721157842036169517, −2.26757546084966269324110456613, −1.66257446756487251353680957291, −0.66037525447870342729129410215,
0.66037525447870342729129410215, 1.66257446756487251353680957291, 2.26757546084966269324110456613, 3.18235696314721157842036169517, 3.43634660745995933925762791811, 3.99415195845520306957084764823, 4.70149803454162952737464687808, 5.21135737120813885125080962772, 5.56250138480305174001193584134, 6.11277041848327263557342668998, 6.61564020931240255898157210391, 7.04066013021810195506913088820, 7.41340325354197965108863025841, 8.009178615776485282232066036514, 8.278659456837051791880753430110
