| L(s) = 1 | + 4-s − 2·5-s − 6·9-s + 16-s + 8·19-s − 2·20-s − 25-s + 4·29-s − 2·31-s − 6·36-s − 12·41-s + 12·45-s − 14·49-s − 24·59-s − 12·61-s + 64-s + 16·71-s + 8·76-s − 16·79-s − 2·80-s + 27·81-s − 12·89-s − 16·95-s − 100-s + 28·101-s − 4·109-s + 4·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 2·9-s + 1/4·16-s + 1.83·19-s − 0.447·20-s − 1/5·25-s + 0.742·29-s − 0.359·31-s − 36-s − 1.87·41-s + 1.78·45-s − 2·49-s − 3.12·59-s − 1.53·61-s + 1/8·64-s + 1.89·71-s + 0.917·76-s − 1.80·79-s − 0.223·80-s + 3·81-s − 1.27·89-s − 1.64·95-s − 0.0999·100-s + 2.78·101-s − 0.383·109-s + 0.371·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.056498267458590117851162621111, −9.024758063663975883170788252068, −8.211267180748869747116136469620, −7.76857292337562945056886541715, −7.68093503876904983414366404311, −6.74316601143042395037310474938, −6.32858579420662158347624019611, −5.75337536396212097143601099281, −5.15554787432694830495821906367, −4.74203912193695697715499891246, −3.64857119618832936893402436683, −3.16188554299090604324575557253, −2.82631239396687231926885349648, −1.57727173319487224423517628693, 0,
1.57727173319487224423517628693, 2.82631239396687231926885349648, 3.16188554299090604324575557253, 3.64857119618832936893402436683, 4.74203912193695697715499891246, 5.15554787432694830495821906367, 5.75337536396212097143601099281, 6.32858579420662158347624019611, 6.74316601143042395037310474938, 7.68093503876904983414366404311, 7.76857292337562945056886541715, 8.211267180748869747116136469620, 9.024758063663975883170788252068, 9.056498267458590117851162621111
