| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·9-s − 14-s + 16-s − 2·17-s − 2·18-s + 4·23-s + 2·25-s + 28-s + 4·31-s − 32-s + 2·34-s + 2·36-s − 2·41-s − 4·46-s + 12·47-s + 49-s − 2·50-s − 56-s − 4·62-s + 2·63-s + 64-s − 2·68-s + 20·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 2/3·9-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.834·23-s + 2/5·25-s + 0.188·28-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 1/3·36-s − 0.312·41-s − 0.589·46-s + 1.75·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.508·62-s + 0.251·63-s + 1/8·64-s − 0.242·68-s + 2.37·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.066421746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066421746\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.19274505607597439649156228478, −9.677108046192235870246199356106, −9.194313781989916405496105062762, −8.528800373646140821271103303358, −8.386038696683392022429998649209, −7.39177891759976496316020420880, −7.28066803334605450056550690805, −6.60343904133397295855360218542, −6.00252687609791041703529005071, −5.25027462301996463773786138541, −4.61343572082246267383198569105, −3.96305578083704870704442222086, −3.00325091538984583711994775428, −2.18055419937459732379532815042, −1.10715236233253586523024540681,
1.10715236233253586523024540681, 2.18055419937459732379532815042, 3.00325091538984583711994775428, 3.96305578083704870704442222086, 4.61343572082246267383198569105, 5.25027462301996463773786138541, 6.00252687609791041703529005071, 6.60343904133397295855360218542, 7.28066803334605450056550690805, 7.39177891759976496316020420880, 8.386038696683392022429998649209, 8.528800373646140821271103303358, 9.194313781989916405496105062762, 9.677108046192235870246199356106, 10.19274505607597439649156228478
