| L(s) = 1 | + 2·2-s − 6·3-s + 2·4-s − 12·6-s + 6·7-s + 18·9-s + 24·11-s − 12·12-s + 24·13-s + 12·14-s − 4·16-s − 24·17-s + 36·18-s − 36·21-s + 48·22-s − 6·23-s + 48·26-s − 54·27-s + 12·28-s − 16·31-s − 8·32-s − 144·33-s − 48·34-s + 36·36-s + 96·37-s − 144·39-s − 96·41-s + ⋯ |
| L(s) = 1 | + 2-s − 2·3-s + 1/2·4-s − 2·6-s + 6/7·7-s + 2·9-s + 2.18·11-s − 12-s + 1.84·13-s + 6/7·14-s − 1/4·16-s − 1.41·17-s + 2·18-s − 1.71·21-s + 2.18·22-s − 0.260·23-s + 1.84·26-s − 2·27-s + 3/7·28-s − 0.516·31-s − 1/4·32-s − 4.36·33-s − 1.41·34-s + 36-s + 2.59·37-s − 3.69·39-s − 2.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.280179325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280179325\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96 T + 4608 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10882 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 186 T + 17298 T^{2} + 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14942 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−15.65081437613277756070886873702, −14.85675972353831863899237310532, −14.66072378348971495484848169715, −13.57085292982982797615314154746, −13.49346587648384673340222799839, −12.63517194630713297048588129221, −11.94484439296439243756063310825, −11.46712333871302629580932963337, −11.21824167426596634666356975027, −10.98086827193740151289353130432, −9.822762371023839079051358572040, −8.997678127128207322577605709863, −8.322712636911506303332342674312, −7.01340998185642322714023690803, −6.35504837536380960253135998046, −6.08538527085902077423606947587, −5.30158693937804148156623983341, −4.29252318897959727144555749919, −3.95109862782897319960374742534, −1.42504353568295107066098519823,
1.42504353568295107066098519823, 3.95109862782897319960374742534, 4.29252318897959727144555749919, 5.30158693937804148156623983341, 6.08538527085902077423606947587, 6.35504837536380960253135998046, 7.01340998185642322714023690803, 8.322712636911506303332342674312, 8.997678127128207322577605709863, 9.822762371023839079051358572040, 10.98086827193740151289353130432, 11.21824167426596634666356975027, 11.46712333871302629580932963337, 11.94484439296439243756063310825, 12.63517194630713297048588129221, 13.49346587648384673340222799839, 13.57085292982982797615314154746, 14.66072378348971495484848169715, 14.85675972353831863899237310532, 15.65081437613277756070886873702
