| L(s) = 1 | + 2·7-s − 9-s − 8·17-s − 4·23-s − 6·25-s + 24·47-s + 3·49-s − 2·63-s − 28·71-s + 4·73-s + 24·79-s + 81-s − 4·97-s + 32·103-s + 20·113-s − 16·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s − 8·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1/3·9-s − 1.94·17-s − 0.834·23-s − 6/5·25-s + 3.50·47-s + 3/7·49-s − 0.251·63-s − 3.32·71-s + 0.468·73-s + 2.70·79-s + 1/9·81-s − 0.406·97-s + 3.15·103-s + 1.88·113-s − 1.46·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.646·153-s + 0.0798·157-s − 0.630·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.891364586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.891364586\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.704198244351956100228088339623, −7.81443443276339300719238693457, −7.76228116116169696420509808612, −7.26484866833276606141366184226, −7.07428814083088902579769386997, −6.57842544119891672534374411653, −5.93737545578447518669596129529, −5.92592270945116938395619895548, −5.71780914359076047139026528655, −4.84058802234822855581330556951, −4.79050865531242017673957410143, −4.26506201785086260545605186343, −4.05908848732914229216413967581, −3.53955306345021236489970265512, −3.05709180159616036834207797884, −2.37451747535111190339194820683, −2.09098063759233980030122446166, −1.91694476182441970239571953197, −0.992391089966167412835417857281, −0.40071296808257268154157009603,
0.40071296808257268154157009603, 0.992391089966167412835417857281, 1.91694476182441970239571953197, 2.09098063759233980030122446166, 2.37451747535111190339194820683, 3.05709180159616036834207797884, 3.53955306345021236489970265512, 4.05908848732914229216413967581, 4.26506201785086260545605186343, 4.79050865531242017673957410143, 4.84058802234822855581330556951, 5.71780914359076047139026528655, 5.92592270945116938395619895548, 5.93737545578447518669596129529, 6.57842544119891672534374411653, 7.07428814083088902579769386997, 7.26484866833276606141366184226, 7.76228116116169696420509808612, 7.81443443276339300719238693457, 8.704198244351956100228088339623
