| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·9-s + 8·13-s + 14-s + 16-s + 2·18-s − 10·25-s − 8·26-s − 28-s + 8·31-s − 32-s − 2·36-s + 16·43-s + 24·47-s + 49-s + 10·50-s + 8·52-s + 56-s − 16·61-s − 8·62-s + 2·63-s + 64-s − 8·67-s + 2·72-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 2.21·13-s + 0.267·14-s + 1/4·16-s + 0.471·18-s − 2·25-s − 1.56·26-s − 0.188·28-s + 1.43·31-s − 0.176·32-s − 1/3·36-s + 2.43·43-s + 3.50·47-s + 1/7·49-s + 1.41·50-s + 1.10·52-s + 0.133·56-s − 2.04·61-s − 1.01·62-s + 0.251·63-s + 1/8·64-s − 0.977·67-s + 0.235·72-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
| \(L(1)\) |
\(\approx\) |
\(0.9391169980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9391169980\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.39623059414136659793705857031, −9.479912708108790860766004869191, −9.205467104844532229657218112988, −8.624667079434487291694904422005, −8.328602349022309226048251750698, −7.55563465371540871894331307120, −7.29200576171944643329126204500, −6.12986038892916091786195943044, −6.04992362572003306544038558924, −5.73625972355331222443538857924, −4.39126934619586232082133319537, −3.87985229263682105648190826671, −3.09628357761679869111062402047, −2.25140513775369021989127014661, −1.00403210582709168291350506092,
1.00403210582709168291350506092, 2.25140513775369021989127014661, 3.09628357761679869111062402047, 3.87985229263682105648190826671, 4.39126934619586232082133319537, 5.73625972355331222443538857924, 6.04992362572003306544038558924, 6.12986038892916091786195943044, 7.29200576171944643329126204500, 7.55563465371540871894331307120, 8.328602349022309226048251750698, 8.624667079434487291694904422005, 9.205467104844532229657218112988, 9.479912708108790860766004869191, 10.39623059414136659793705857031
