| L(s) = 1 | − 2·3-s − 3·4-s + 3·9-s + 6·12-s − 2·13-s + 5·16-s − 4·17-s + 16·23-s − 6·25-s − 4·27-s − 12·29-s − 9·36-s + 4·39-s − 10·48-s + 2·49-s + 8·51-s + 6·52-s + 12·53-s + 12·61-s − 3·64-s + 12·68-s − 32·69-s + 12·75-s − 8·79-s + 5·81-s + 24·87-s − 48·92-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 9-s + 1.73·12-s − 0.554·13-s + 5/4·16-s − 0.970·17-s + 3.33·23-s − 6/5·25-s − 0.769·27-s − 2.22·29-s − 3/2·36-s + 0.640·39-s − 1.44·48-s + 2/7·49-s + 1.12·51-s + 0.832·52-s + 1.64·53-s + 1.53·61-s − 3/8·64-s + 1.45·68-s − 3.85·69-s + 1.38·75-s − 0.900·79-s + 5/9·81-s + 2.57·87-s − 5.00·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.975261543279013968212899152873, −8.768985116389244790636560898392, −7.82831259030341079249655300429, −7.46995816184461358823952634116, −6.87625566807413692768701148012, −6.57112145512025234345666775516, −5.59092668167682561675172765450, −5.31622482563829752699303340190, −5.10830147517302686718916077334, −4.20407903283329927801590362766, −4.11432405402007502173846810332, −3.20967362704724919819221481375, −2.18490783609343058155048362276, −1.01190903051797145927138124185, 0,
1.01190903051797145927138124185, 2.18490783609343058155048362276, 3.20967362704724919819221481375, 4.11432405402007502173846810332, 4.20407903283329927801590362766, 5.10830147517302686718916077334, 5.31622482563829752699303340190, 5.59092668167682561675172765450, 6.57112145512025234345666775516, 6.87625566807413692768701148012, 7.46995816184461358823952634116, 7.82831259030341079249655300429, 8.768985116389244790636560898392, 8.975261543279013968212899152873
