| L(s) = 1 | − 2-s + 2·3-s + 2·5-s − 2·6-s + 8-s + 3·9-s − 2·10-s − 11-s + 4·13-s + 4·15-s − 16-s − 3·18-s + 2·19-s + 22-s + 2·24-s + 5·25-s − 4·26-s + 10·27-s + 12·29-s − 4·30-s − 4·31-s − 2·33-s − 2·37-s − 2·38-s + 8·39-s + 2·40-s + 16·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 9-s − 0.632·10-s − 0.301·11-s + 1.10·13-s + 1.03·15-s − 1/4·16-s − 0.707·18-s + 0.458·19-s + 0.213·22-s + 0.408·24-s + 25-s − 0.784·26-s + 1.92·27-s + 2.22·29-s − 0.730·30-s − 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.324·38-s + 1.28·39-s + 0.316·40-s + 2.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.333937939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.333937939\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−9.878480197869522918873785902868, −9.569132257440736583791078470820, −9.277313174607398447291313851018, −8.833126940553592347636614710483, −8.528670673808767553411287838081, −8.185673751954384129227300022030, −7.78695518664863755018312269860, −7.29904018668731221920527153075, −6.70386142176735825779759230728, −6.58148929550446217701169107224, −5.68212035210045430791930913610, −5.63999097904389062575008372897, −4.80590847402103469607571089897, −4.21956228880155036300121981627, −4.02676480271910917857165975411, −2.98330363049331644095271697287, −2.79555753753060316965791167108, −2.30660920713042264604131737042, −1.21874606892508270594140318378, −1.10051110655016347381152921607,
1.10051110655016347381152921607, 1.21874606892508270594140318378, 2.30660920713042264604131737042, 2.79555753753060316965791167108, 2.98330363049331644095271697287, 4.02676480271910917857165975411, 4.21956228880155036300121981627, 4.80590847402103469607571089897, 5.63999097904389062575008372897, 5.68212035210045430791930913610, 6.58148929550446217701169107224, 6.70386142176735825779759230728, 7.29904018668731221920527153075, 7.78695518664863755018312269860, 8.185673751954384129227300022030, 8.528670673808767553411287838081, 8.833126940553592347636614710483, 9.277313174607398447291313851018, 9.569132257440736583791078470820, 9.878480197869522918873785902868
