| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 9-s − 2·10-s + 13-s + 16-s + 18-s + 2·20-s − 25-s − 26-s + 14·29-s − 32-s − 36-s + 14·37-s − 2·40-s + 8·41-s − 2·45-s + 9·49-s + 50-s + 52-s − 4·53-s − 14·58-s + 4·61-s + 64-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 1/3·9-s − 0.632·10-s + 0.277·13-s + 1/4·16-s + 0.235·18-s + 0.447·20-s − 1/5·25-s − 0.196·26-s + 2.59·29-s − 0.176·32-s − 1/6·36-s + 2.30·37-s − 0.316·40-s + 1.24·41-s − 0.298·45-s + 9/7·49-s + 0.141·50-s + 0.138·52-s − 0.549·53-s − 1.83·58-s + 0.512·61-s + 1/8·64-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.977312778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977312778\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−7.968524233946697847762605196727, −7.80523321874401660046491367151, −7.11853937618382367033690194995, −6.71352017423295661998284998666, −6.23832577849018197974029663600, −5.89227290181790023043969529447, −5.65154724896199128000363529863, −4.82190555880230931711220225865, −4.50119107836766960602317785734, −3.87338505534032842973611046811, −3.13136137098566183715553529307, −2.50734690717421409848107112046, −2.35272847404389910766238459806, −1.31263696105008514097478225671, −0.77682140340402947244140202237,
0.77682140340402947244140202237, 1.31263696105008514097478225671, 2.35272847404389910766238459806, 2.50734690717421409848107112046, 3.13136137098566183715553529307, 3.87338505534032842973611046811, 4.50119107836766960602317785734, 4.82190555880230931711220225865, 5.65154724896199128000363529863, 5.89227290181790023043969529447, 6.23832577849018197974029663600, 6.71352017423295661998284998666, 7.11853937618382367033690194995, 7.80523321874401660046491367151, 7.968524233946697847762605196727
