| L(s) = 1 | − 2-s − 3-s + 6-s − 7-s + 8-s − 3·11-s − 7·13-s + 14-s − 16-s − 3·17-s − 5·19-s + 21-s + 3·22-s − 6·23-s − 24-s − 10·25-s + 7·26-s + 27-s + 3·29-s − 8·31-s + 3·33-s + 3·34-s + 4·37-s + 5·38-s + 7·39-s + 3·41-s − 42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.904·11-s − 1.94·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s − 1.14·19-s + 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s − 2·25-s + 1.37·26-s + 0.192·27-s + 0.557·29-s − 1.43·31-s + 0.522·33-s + 0.514·34-s + 0.657·37-s + 0.811·38-s + 1.12·39-s + 0.468·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.37202673827620200804606497013, −10.23495964714052628902466770313, −9.651121492418243418918015037962, −9.453642667954677657666806461203, −8.897255014354849941293596911901, −8.317257046262649002426373369172, −7.83873337026315190808385447530, −7.55815905293563325856043959559, −7.09243760743157993969694933263, −6.50536205682837573880549311290, −5.87080297661846121160247021845, −5.66492534122420757550052588777, −4.92195010975224217259434907813, −4.37589542230758019747744243792, −4.05536305569503541777473301077, −3.00578576155251164584655802355, −2.30054975701406865933678141656, −1.87798864913775327366154204737, 0, 0,
1.87798864913775327366154204737, 2.30054975701406865933678141656, 3.00578576155251164584655802355, 4.05536305569503541777473301077, 4.37589542230758019747744243792, 4.92195010975224217259434907813, 5.66492534122420757550052588777, 5.87080297661846121160247021845, 6.50536205682837573880549311290, 7.09243760743157993969694933263, 7.55815905293563325856043959559, 7.83873337026315190808385447530, 8.317257046262649002426373369172, 8.897255014354849941293596911901, 9.453642667954677657666806461203, 9.651121492418243418918015037962, 10.23495964714052628902466770313, 10.37202673827620200804606497013
