| L(s) = 1 | + 2·5-s − 9-s + 3·25-s + 4·29-s − 2·45-s − 2·49-s + 81-s − 4·101-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·5-s − 9-s + 3·25-s + 4·29-s − 2·45-s − 2·49-s + 81-s − 4·101-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.787999785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.787999785\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.722211238201394823824434273095, −9.158699775033595978465826499023, −8.729735099517333899856591114411, −8.658403263914324548882351756993, −8.007443160996929553136465974534, −7.916606579260750959585291395642, −6.88572080223347908746394241417, −6.77213597953916518965291958308, −6.27471380149531546630502330110, −6.21026124085781930824898870166, −5.58909745830828812005455090872, −5.15471077492020672379834550451, −4.88752387105372933368582222530, −4.49269003201539288075684986284, −3.69336988928310007198067120277, −2.97311842504770788806316327359, −2.69067008155724674189853371907, −2.45206276354445370766216181844, −1.52671242857185594892421627128, −1.11462813380787190878932701799,
1.11462813380787190878932701799, 1.52671242857185594892421627128, 2.45206276354445370766216181844, 2.69067008155724674189853371907, 2.97311842504770788806316327359, 3.69336988928310007198067120277, 4.49269003201539288075684986284, 4.88752387105372933368582222530, 5.15471077492020672379834550451, 5.58909745830828812005455090872, 6.21026124085781930824898870166, 6.27471380149531546630502330110, 6.77213597953916518965291958308, 6.88572080223347908746394241417, 7.916606579260750959585291395642, 8.007443160996929553136465974534, 8.658403263914324548882351756993, 8.729735099517333899856591114411, 9.158699775033595978465826499023, 9.722211238201394823824434273095
