| L(s) = 1 | + 2·3-s − 4-s + 2·7-s + 3·9-s − 2·12-s + 16-s − 2·17-s + 6·19-s + 4·21-s + 16·23-s + 4·27-s − 2·28-s − 3·36-s + 6·37-s + 2·48-s − 11·49-s − 4·51-s + 12·57-s + 16·59-s + 6·63-s − 64-s + 2·68-s + 32·69-s + 8·73-s − 6·76-s + 5·81-s − 4·84-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.755·7-s + 9-s − 0.577·12-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.872·21-s + 3.33·23-s + 0.769·27-s − 0.377·28-s − 1/2·36-s + 0.986·37-s + 0.288·48-s − 1.57·49-s − 0.560·51-s + 1.58·57-s + 2.08·59-s + 0.755·63-s − 1/8·64-s + 0.242·68-s + 3.85·69-s + 0.936·73-s − 0.688·76-s + 5/9·81-s − 0.436·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.392710757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.392710757\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−8.953437266547120230324070130189, −8.775038061632553088223337416682, −8.417509340731736207069433502406, −8.012896242087041538564959686228, −7.54880436723633880549307497293, −7.37034497738504992623997348425, −6.88917404654469952423559197959, −6.62221251103740744320377410909, −5.93949519215987153309427941768, −5.46151443502371642760214485258, −4.92105947126811246238590867589, −4.77784645658636268850378098354, −4.49267728916769438680480260447, −3.67583275624564914238352384063, −3.26642484880613977787182075083, −3.15008615969782863125410887926, −2.37226175374957258653872878470, −2.00348976563689881533517063952, −1.04396355304095248302138119646, −0.936087930532685525070541061256,
0.936087930532685525070541061256, 1.04396355304095248302138119646, 2.00348976563689881533517063952, 2.37226175374957258653872878470, 3.15008615969782863125410887926, 3.26642484880613977787182075083, 3.67583275624564914238352384063, 4.49267728916769438680480260447, 4.77784645658636268850378098354, 4.92105947126811246238590867589, 5.46151443502371642760214485258, 5.93949519215987153309427941768, 6.62221251103740744320377410909, 6.88917404654469952423559197959, 7.37034497738504992623997348425, 7.54880436723633880549307497293, 8.012896242087041538564959686228, 8.417509340731736207069433502406, 8.775038061632553088223337416682, 8.953437266547120230324070130189
