| L(s) = 1 | − 2·4-s − 8·7-s + 9-s + 4·16-s − 2·17-s + 18·23-s − 25-s + 16·28-s + 4·31-s − 2·36-s − 6·41-s − 12·47-s + 34·49-s − 8·63-s − 8·64-s + 4·68-s + 24·71-s + 4·73-s − 20·79-s + 81-s − 36·92-s − 32·97-s + 2·100-s + 10·103-s − 32·112-s − 18·113-s + 16·119-s + ⋯ |
| L(s) = 1 | − 4-s − 3.02·7-s + 1/3·9-s + 16-s − 0.485·17-s + 3.75·23-s − 1/5·25-s + 3.02·28-s + 0.718·31-s − 1/3·36-s − 0.937·41-s − 1.75·47-s + 34/7·49-s − 1.00·63-s − 64-s + 0.485·68-s + 2.84·71-s + 0.468·73-s − 2.25·79-s + 1/9·81-s − 3.75·92-s − 3.24·97-s + 1/5·100-s + 0.985·103-s − 3.02·112-s − 1.69·113-s + 1.46·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−9.250915914068804964862203626670, −8.618966621189813626434174913816, −8.228682176903090380178701350148, −7.31936717375093530826366945214, −6.80299500068960285512965295533, −6.66463575826783700250676263362, −6.19889841643513433803233134244, −5.21243235840224221221520663568, −5.14306131647545403565477997005, −4.18993695410695019989623309599, −3.63699123933177341557225053751, −3.00632648437536992753290178800, −2.85877051829134286082199945018, −1.08612370698107395863808052921, 0,
1.08612370698107395863808052921, 2.85877051829134286082199945018, 3.00632648437536992753290178800, 3.63699123933177341557225053751, 4.18993695410695019989623309599, 5.14306131647545403565477997005, 5.21243235840224221221520663568, 6.19889841643513433803233134244, 6.66463575826783700250676263362, 6.80299500068960285512965295533, 7.31936717375093530826366945214, 8.228682176903090380178701350148, 8.618966621189813626434174913816, 9.250915914068804964862203626670
