| L(s) = 1 | + 4-s + 6·9-s + 11-s + 16-s + 16·23-s + 2·25-s + 6·36-s − 12·37-s + 44-s − 7·49-s − 12·53-s + 64-s + 8·67-s + 27·81-s + 16·92-s + 6·99-s + 2·100-s − 12·113-s + 121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s − 12·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2·9-s + 0.301·11-s + 1/4·16-s + 3.33·23-s + 2/5·25-s + 36-s − 1.97·37-s + 0.150·44-s − 49-s − 1.64·53-s + 1/8·64-s + 0.977·67-s + 3·81-s + 1.66·92-s + 0.603·99-s + 1/5·100-s − 1.12·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s − 0.986·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.665622171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665622171\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
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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
−9.057391921212177938213328394364, −8.494149622254059818080035076185, −7.85272986534120038415407467632, −7.32615693509824401196563041856, −7.03037470409764709156473949624, −6.66654967118821035672744944634, −6.33539770631464590838934287146, −5.29637285243714588581784021659, −4.95407749784441441983128839476, −4.60740616023990205481710981393, −3.71793124530211373357479879031, −3.34620467406033722077754124678, −2.57866198506403861623716189696, −1.58133106765564568625613031282, −1.16708884817637229457185444327,
1.16708884817637229457185444327, 1.58133106765564568625613031282, 2.57866198506403861623716189696, 3.34620467406033722077754124678, 3.71793124530211373357479879031, 4.60740616023990205481710981393, 4.95407749784441441983128839476, 5.29637285243714588581784021659, 6.33539770631464590838934287146, 6.66654967118821035672744944634, 7.03037470409764709156473949624, 7.32615693509824401196563041856, 7.85272986534120038415407467632, 8.494149622254059818080035076185, 9.057391921212177938213328394364
