| L(s) = 1 | + 2·2-s + 12·5-s + 7·7-s − 8·8-s + 24·10-s + 60·11-s + 79·13-s + 14·14-s − 16·16-s + 216·17-s + 22·19-s + 120·22-s − 132·23-s + 125·25-s + 158·26-s + 96·29-s − 20·31-s + 432·34-s + 84·35-s − 338·37-s + 44·38-s − 96·40-s + 192·41-s − 488·43-s − 264·46-s + 204·47-s + 343·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.07·5-s + 0.377·7-s − 0.353·8-s + 0.758·10-s + 1.64·11-s + 1.68·13-s + 0.267·14-s − 1/4·16-s + 3.08·17-s + 0.265·19-s + 1.16·22-s − 1.19·23-s + 25-s + 1.19·26-s + 0.614·29-s − 0.115·31-s + 2.17·34-s + 0.405·35-s − 1.50·37-s + 0.187·38-s − 0.379·40-s + 0.731·41-s − 1.73·43-s − 0.846·46-s + 0.633·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.503243760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.503243760\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 12 T + 19 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + 4 p T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 60 T + 2269 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 79 T + 4044 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 132 T + 5257 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 96 T - 15173 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T - 29391 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 169 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 192 T - 32057 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 488 T + 158637 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 204 T - 62207 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 156 T - 181043 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 83 T - 220092 T^{2} + 83 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 47 T - 298554 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 216 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 p T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 529 T - 213198 T^{2} - 529 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1128 T + 700597 T^{2} + 1128 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 605 T - 546648 T^{2} + 605 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.53391227167221971680756509917, −12.05292311384642281798719194442, −12.00637431200768342593986901282, −11.29734739163912714133676392045, −10.48633144660124869733949125365, −10.27037665441606254743864097376, −9.471793241339961422436437154859, −9.330536004402598772743890940617, −8.315809101436796690448293729717, −8.297713369193319960972053123942, −7.26130583536533061184518853627, −6.61518017226888555513007221944, −5.93153466411979294952631010759, −5.76052544183968496914946897613, −5.12016648520626914993888084680, −4.14256552243559666128695948512, −3.59868446260258285242094761035, −3.02959564411468035959265649750, −1.43298413710066483699653885994, −1.33330507053561638362858929874,
1.33330507053561638362858929874, 1.43298413710066483699653885994, 3.02959564411468035959265649750, 3.59868446260258285242094761035, 4.14256552243559666128695948512, 5.12016648520626914993888084680, 5.76052544183968496914946897613, 5.93153466411979294952631010759, 6.61518017226888555513007221944, 7.26130583536533061184518853627, 8.297713369193319960972053123942, 8.315809101436796690448293729717, 9.330536004402598772743890940617, 9.471793241339961422436437154859, 10.27037665441606254743864097376, 10.48633144660124869733949125365, 11.29734739163912714133676392045, 12.00637431200768342593986901282, 12.05292311384642281798719194442, 12.53391227167221971680756509917
