| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 6·11-s − 15-s − 2·21-s + 25-s + 27-s + 6·33-s + 2·35-s − 8·43-s − 45-s − 2·49-s + 12·53-s − 6·55-s + 18·59-s + 4·61-s − 2·63-s + 4·67-s + 12·71-s + 75-s − 12·77-s + 81-s + 6·99-s + 10·103-s + 2·105-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.258·15-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.04·33-s + 0.338·35-s − 1.21·43-s − 0.149·45-s − 2/7·49-s + 1.64·53-s − 0.809·55-s + 2.34·59-s + 0.512·61-s − 0.251·63-s + 0.488·67-s + 1.42·71-s + 0.115·75-s − 1.36·77-s + 1/9·81-s + 0.603·99-s + 0.985·103-s + 0.195·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963907807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963907807\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.986817345232143354160192898618, −8.624384126853811739476276912996, −8.222151557355997532915384866309, −7.63588507152931622264611878773, −6.99152317761459520818406697833, −6.66985734205751366789458017246, −6.44138367987963469752015935562, −5.57548968519602015653592828161, −5.08972778796573881835305963752, −4.21232854349724191593823438788, −3.85423103781006743254277566399, −3.48288543200374090452875304293, −2.70474550224093063124939423562, −1.89037268545330142266197998729, −0.893764133731049935751559998534,
0.893764133731049935751559998534, 1.89037268545330142266197998729, 2.70474550224093063124939423562, 3.48288543200374090452875304293, 3.85423103781006743254277566399, 4.21232854349724191593823438788, 5.08972778796573881835305963752, 5.57548968519602015653592828161, 6.44138367987963469752015935562, 6.66985734205751366789458017246, 6.99152317761459520818406697833, 7.63588507152931622264611878773, 8.222151557355997532915384866309, 8.624384126853811739476276912996, 8.986817345232143354160192898618
