| 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 + 9·23-s + 8·25-s − 3·28-s − 6·32-s + 3·36-s + 4·37-s + 7·43-s − 18·46-s − 6·49-s − 16·50-s − 18·53-s + 4·56-s − 63-s + 7·64-s + 15·67-s + 3·71-s − 4·72-s − 8·74-s − 2·79-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 + 1.87·23-s + 8/5·25-s − 0.566·28-s − 1.06·32-s + 1/2·36-s + 0.657·37-s + 1.06·43-s − 2.65·46-s − 6/7·49-s − 2.26·50-s − 2.47·53-s + 0.534·56-s − 0.125·63-s + 7/8·64-s + 1.83·67-s + 0.356·71-s − 0.471·72-s − 0.929·74-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5711183854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5711183854\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 14 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 148 T^{2} + p^{2} T^{4} \) |
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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
−11.06242556734797633316617654325, −10.75142325686499068465966747941, −10.03580089431797300060009499074, −9.496854124180573508165716157921, −9.145978517862082072816079858900, −8.570611933157460167149885055115, −7.946093092912831285360621515242, −7.34391638071145319827988985542, −6.73529519082847451308985885003, −6.38183167189614169029600475754, −5.35998631133052630855741203130, −4.60746508936177210269135914867, −3.34534227552155671660153330050, −2.62780247878860436110860336078, −1.19064202541676050214281881703,
1.19064202541676050214281881703, 2.62780247878860436110860336078, 3.34534227552155671660153330050, 4.60746508936177210269135914867, 5.35998631133052630855741203130, 6.38183167189614169029600475754, 6.73529519082847451308985885003, 7.34391638071145319827988985542, 7.946093092912831285360621515242, 8.570611933157460167149885055115, 9.145978517862082072816079858900, 9.496854124180573508165716157921, 10.03580089431797300060009499074, 10.75142325686499068465966747941, 11.06242556734797633316617654325
