| L(s) = 1 | + 2·2-s + 2·4-s + 6·11-s − 4·16-s − 6·17-s − 2·19-s + 12·22-s − 9·25-s − 8·32-s − 12·34-s − 4·38-s + 16·41-s − 2·43-s + 12·44-s − 5·49-s − 18·50-s − 8·64-s + 16·67-s − 12·68-s − 22·73-s − 4·76-s + 32·82-s − 8·83-s − 4·86-s − 20·89-s − 4·97-s − 10·98-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.80·11-s − 16-s − 1.45·17-s − 0.458·19-s + 2.55·22-s − 9/5·25-s − 1.41·32-s − 2.05·34-s − 0.648·38-s + 2.49·41-s − 0.304·43-s + 1.80·44-s − 5/7·49-s − 2.54·50-s − 64-s + 1.95·67-s − 1.45·68-s − 2.57·73-s − 0.458·76-s + 3.53·82-s − 0.878·83-s − 0.431·86-s − 2.11·89-s − 0.406·97-s − 1.01·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−7.33179869161333338136909211767, −6.96497281907628557036215151736, −6.61173828601165805902581538030, −6.12831923497636436478284774657, −5.91491105383502669780569778800, −5.50429222183328751376839026433, −4.75227634922127063518685270995, −4.25777573246989549492697138862, −4.20883425493377341015603476651, −3.73823129232702180128955205179, −3.13945209799862894261132400983, −2.44031011733442394942262196334, −2.03599320139561910762409600477, −1.27951136452794980291664737706, 0,
1.27951136452794980291664737706, 2.03599320139561910762409600477, 2.44031011733442394942262196334, 3.13945209799862894261132400983, 3.73823129232702180128955205179, 4.20883425493377341015603476651, 4.25777573246989549492697138862, 4.75227634922127063518685270995, 5.50429222183328751376839026433, 5.91491105383502669780569778800, 6.12831923497636436478284774657, 6.61173828601165805902581538030, 6.96497281907628557036215151736, 7.33179869161333338136909211767
