| L(s) = 1 | − 2·3-s + 2·5-s − 9-s − 2·11-s + 2·13-s − 4·15-s − 4·17-s − 12·19-s − 4·23-s − 7·25-s + 6·27-s − 2·29-s + 6·31-s + 4·33-s + 8·37-s − 4·39-s + 8·41-s − 10·43-s − 2·45-s + 2·47-s − 6·49-s + 8·51-s − 2·53-s − 4·55-s + 24·57-s − 4·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.03·15-s − 0.970·17-s − 2.75·19-s − 0.834·23-s − 7/5·25-s + 1.15·27-s − 0.371·29-s + 1.07·31-s + 0.696·33-s + 1.31·37-s − 0.640·39-s + 1.24·41-s − 1.52·43-s − 0.298·45-s + 0.291·47-s − 6/7·49-s + 1.12·51-s − 0.274·53-s − 0.539·55-s + 3.17·57-s − 0.520·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 77 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.935108489628942822524308248902, −8.752224356421627575261631364288, −8.109341981290463959091062380491, −8.098372663257254115582378366429, −7.52474121636763809768949184922, −6.76966144069386946384857676520, −6.35496258510786618587539178043, −6.26595221991533593513321338243, −5.88224680086553034473616615342, −5.68489733783248772871386675851, −4.90316089408612750083044511199, −4.73518046796162665483122124709, −3.96437482878671976976643338134, −3.94689135092261173088975092856, −2.83904810850298681045805294085, −2.39509171048494079004273609655, −2.06898729966108955556495974338, −1.33705204080188680039538287194, 0, 0,
1.33705204080188680039538287194, 2.06898729966108955556495974338, 2.39509171048494079004273609655, 2.83904810850298681045805294085, 3.94689135092261173088975092856, 3.96437482878671976976643338134, 4.73518046796162665483122124709, 4.90316089408612750083044511199, 5.68489733783248772871386675851, 5.88224680086553034473616615342, 6.26595221991533593513321338243, 6.35496258510786618587539178043, 6.76966144069386946384857676520, 7.52474121636763809768949184922, 8.098372663257254115582378366429, 8.109341981290463959091062380491, 8.752224356421627575261631364288, 8.935108489628942822524308248902
