| L(s) = 1 | − 9-s − 12·29-s + 8·31-s + 20·41-s − 49-s − 8·59-s + 12·61-s + 24·71-s + 81-s − 20·89-s + 12·101-s + 36·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.22·29-s + 1.43·31-s + 3.12·41-s − 1/7·49-s − 1.04·59-s + 1.53·61-s + 2.84·71-s + 1/9·81-s − 2.11·89-s + 1.19·101-s + 3.44·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.619655888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.619655888\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.388946152240857539446848149231, −8.311564171872903149823521412204, −7.72692557635064699284544019937, −7.69741612559629214971870706200, −7.02647367175294292688542100664, −6.91801813754533825957006981227, −6.29075055498796558623863832790, −6.01178395616630732213600751769, −5.61628620490480239744862699579, −5.40899968721193656433874111627, −4.76303449645869002579281427108, −4.49436431549070910671993091140, −3.87298290411258498187314470490, −3.81827441082834718641588022777, −3.01323098422640865760285755733, −2.82100696871792235006285885891, −2.07188291705492064489318334839, −1.92998342119513618808918746058, −0.945574834503128380824325450047, −0.55530145918170494968855855263,
0.55530145918170494968855855263, 0.945574834503128380824325450047, 1.92998342119513618808918746058, 2.07188291705492064489318334839, 2.82100696871792235006285885891, 3.01323098422640865760285755733, 3.81827441082834718641588022777, 3.87298290411258498187314470490, 4.49436431549070910671993091140, 4.76303449645869002579281427108, 5.40899968721193656433874111627, 5.61628620490480239744862699579, 6.01178395616630732213600751769, 6.29075055498796558623863832790, 6.91801813754533825957006981227, 7.02647367175294292688542100664, 7.69741612559629214971870706200, 7.72692557635064699284544019937, 8.311564171872903149823521412204, 8.388946152240857539446848149231
