| L(s) = 1 | + 4·2-s + 8·4-s + 2·5-s + 4·7-s + 8·8-s + 8·10-s + 2·11-s − 13-s + 16·14-s − 4·16-s − 19-s + 16·20-s + 8·22-s + 5·25-s − 4·26-s + 32·28-s − 4·29-s + 18·31-s − 32·32-s + 8·35-s − 3·37-s − 4·38-s + 16·40-s + 10·41-s − 5·43-s + 16·44-s − 12·47-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 0.894·5-s + 1.51·7-s + 2.82·8-s + 2.52·10-s + 0.603·11-s − 0.277·13-s + 4.27·14-s − 16-s − 0.229·19-s + 3.57·20-s + 1.70·22-s + 25-s − 0.784·26-s + 6.04·28-s − 0.742·29-s + 3.23·31-s − 5.65·32-s + 1.35·35-s − 0.493·37-s − 0.648·38-s + 2.52·40-s + 1.56·41-s − 0.762·43-s + 2.41·44-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.62623328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.62623328\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.19053916535044429935864695680, −10.97059778215150589588751946496, −10.16404022956856731541099034511, −9.839307603528317056759673567786, −9.052661380011445523997421593135, −8.935372496729373069899333276050, −8.094599893040812520943863157413, −7.85606869987153944273749469748, −6.78104534233584435121883601479, −6.75246565263516057586962859042, −6.09460906317154729699380881200, −5.78928086463902417604200726998, −5.23965135611607732885610810755, −4.76743981350019007626894731994, −4.42244810487173527925568776460, −4.26554808333242869941222470969, −3.08992101722162084092661256947, −3.01879187002628095715612568598, −2.12973441212399148289647383768, −1.49505257794488067939488039762,
1.49505257794488067939488039762, 2.12973441212399148289647383768, 3.01879187002628095715612568598, 3.08992101722162084092661256947, 4.26554808333242869941222470969, 4.42244810487173527925568776460, 4.76743981350019007626894731994, 5.23965135611607732885610810755, 5.78928086463902417604200726998, 6.09460906317154729699380881200, 6.75246565263516057586962859042, 6.78104534233584435121883601479, 7.85606869987153944273749469748, 8.094599893040812520943863157413, 8.935372496729373069899333276050, 9.052661380011445523997421593135, 9.839307603528317056759673567786, 10.16404022956856731541099034511, 10.97059778215150589588751946496, 11.19053916535044429935864695680
