| L(s) = 1 | − 2·3-s + 4-s + 3·9-s − 11-s − 2·12-s + 16-s − 8·23-s − 10·25-s − 4·27-s + 12·31-s + 2·33-s + 3·36-s − 12·37-s − 44-s + 4·47-s − 2·48-s + 49-s + 12·53-s − 16·59-s + 64-s − 8·67-s + 16·69-s + 20·75-s + 5·81-s + 20·89-s − 8·92-s − 24·93-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 9-s − 0.301·11-s − 0.577·12-s + 1/4·16-s − 1.66·23-s − 2·25-s − 0.769·27-s + 2.15·31-s + 0.348·33-s + 1/2·36-s − 1.97·37-s − 0.150·44-s + 0.583·47-s − 0.288·48-s + 1/7·49-s + 1.64·53-s − 2.08·59-s + 1/8·64-s − 0.977·67-s + 1.92·69-s + 2.30·75-s + 5/9·81-s + 2.11·89-s − 0.834·92-s − 2.48·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.50195258614813863797879007663, −7.01534336846329507543780590242, −6.42489977862829018225919109306, −6.10098520284764516138950755290, −6.00314373517083763968923561059, −5.36900763128811539114773250289, −5.00286409242662365238378343541, −4.40989342728991428553771509734, −4.05576474933775594859751576814, −3.51350398110842506848938122210, −2.90878683357439788919878297904, −1.96324335591924275791437827569, −1.95915855001918838067318802070, −0.866846976089747955444418497930, 0,
0.866846976089747955444418497930, 1.95915855001918838067318802070, 1.96324335591924275791437827569, 2.90878683357439788919878297904, 3.51350398110842506848938122210, 4.05576474933775594859751576814, 4.40989342728991428553771509734, 5.00286409242662365238378343541, 5.36900763128811539114773250289, 6.00314373517083763968923561059, 6.10098520284764516138950755290, 6.42489977862829018225919109306, 7.01534336846329507543780590242, 7.50195258614813863797879007663
