| L(s) = 1 | + 2·3-s + 3·9-s − 4·11-s − 8·19-s + 6·25-s + 4·27-s − 8·33-s − 8·41-s − 16·43-s + 49-s − 16·57-s + 16·59-s + 8·67-s + 12·73-s + 12·75-s + 5·81-s − 24·83-s + 8·89-s − 4·97-s − 12·99-s − 20·107-s + 4·113-s − 10·121-s − 16·123-s + 127-s − 32·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 1.20·11-s − 1.83·19-s + 6/5·25-s + 0.769·27-s − 1.39·33-s − 1.24·41-s − 2.43·43-s + 1/7·49-s − 2.11·57-s + 2.08·59-s + 0.977·67-s + 1.40·73-s + 1.38·75-s + 5/9·81-s − 2.63·83-s + 0.847·89-s − 0.406·97-s − 1.20·99-s − 1.93·107-s + 0.376·113-s − 0.909·121-s − 1.44·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.050306287517390944376390946650, −7.77278723062985727351711147882, −6.85036368281080789512161503853, −6.83715734248313427167981756631, −6.48144389669749178167352152901, −5.44334341616198064453307723167, −5.31505211056085532901675130626, −4.66169216854101764844813417996, −4.21546713870310250114993857360, −3.58625452137859596295479508108, −3.16076221185606682123289148521, −2.45390467337760422022857720016, −2.19264027338821122800426191253, −1.34272176216502313926400283488, 0,
1.34272176216502313926400283488, 2.19264027338821122800426191253, 2.45390467337760422022857720016, 3.16076221185606682123289148521, 3.58625452137859596295479508108, 4.21546713870310250114993857360, 4.66169216854101764844813417996, 5.31505211056085532901675130626, 5.44334341616198064453307723167, 6.48144389669749178167352152901, 6.83715734248313427167981756631, 6.85036368281080789512161503853, 7.77278723062985727351711147882, 8.050306287517390944376390946650
