| L(s) = 1 | − 2·2-s − 4-s + 4·5-s + 8·8-s − 9-s − 8·10-s − 7·16-s + 2·18-s − 4·20-s + 8·23-s + 2·25-s − 14·32-s + 36-s + 20·37-s + 32·40-s + 10·41-s − 16·43-s − 4·45-s − 16·46-s − 2·49-s − 4·50-s − 16·59-s + 12·61-s + 35·64-s − 8·72-s − 20·73-s − 40·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 1.78·5-s + 2.82·8-s − 1/3·9-s − 2.52·10-s − 7/4·16-s + 0.471·18-s − 0.894·20-s + 1.66·23-s + 2/5·25-s − 2.47·32-s + 1/6·36-s + 3.28·37-s + 5.05·40-s + 1.56·41-s − 2.43·43-s − 0.596·45-s − 2.35·46-s − 2/7·49-s − 0.565·50-s − 2.08·59-s + 1.53·61-s + 35/8·64-s − 0.942·72-s − 2.34·73-s − 4.64·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1830609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1830609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9267021169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9267021169\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| 41 | $C_2$ | \( 1 - 10 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−9.667187541238594779027839940294, −9.463582036099241768413304477102, −9.176349570646810901135052600200, −8.674611221712237511154819858234, −8.441127328901310167908637353865, −7.84277654508129065115779359008, −7.63512269106022326947200614596, −7.14645911391892109192668063377, −6.49105242568971150474979460765, −6.11289016728692206051004336535, −5.70418230504966267685290571938, −5.06522733163499222035539324566, −5.01432466585164993597028538689, −4.21439965152871875438722249974, −3.98485215368642713211103007112, −2.88720327647774964118584909950, −2.62039274903634340370048717190, −1.58503377055351588573632660438, −1.44896020291427547031587021848, −0.55457234513428761132377766784,
0.55457234513428761132377766784, 1.44896020291427547031587021848, 1.58503377055351588573632660438, 2.62039274903634340370048717190, 2.88720327647774964118584909950, 3.98485215368642713211103007112, 4.21439965152871875438722249974, 5.01432466585164993597028538689, 5.06522733163499222035539324566, 5.70418230504966267685290571938, 6.11289016728692206051004336535, 6.49105242568971150474979460765, 7.14645911391892109192668063377, 7.63512269106022326947200614596, 7.84277654508129065115779359008, 8.441127328901310167908637353865, 8.674611221712237511154819858234, 9.176349570646810901135052600200, 9.463582036099241768413304477102, 9.667187541238594779027839940294
