| L(s) = 1 | + 3-s + 2·4-s + 9-s + 11-s + 2·12-s − 10·23-s + 25-s + 27-s − 5·31-s + 33-s + 2·36-s − 2·37-s + 2·44-s − 8·47-s + 49-s + 4·59-s − 8·64-s − 8·67-s − 10·69-s + 18·71-s + 75-s + 81-s − 14·89-s − 20·92-s − 5·93-s + 14·97-s + 99-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 2.08·23-s + 1/5·25-s + 0.192·27-s − 0.898·31-s + 0.174·33-s + 1/3·36-s − 0.328·37-s + 0.301·44-s − 1.16·47-s + 1/7·49-s + 0.520·59-s − 64-s − 0.977·67-s − 1.20·69-s + 2.13·71-s + 0.115·75-s + 1/9·81-s − 1.48·89-s − 2.08·92-s − 0.518·93-s + 1.42·97-s + 0.100·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.30661797541290821455696261649, −7.03189212108301260064437659817, −6.63216653749155622423965381578, −6.23404778308086977014137472094, −5.79202265149903668260362805735, −5.40844333678785302124910491259, −4.70329769108401126404570380961, −4.34563034217913995952762795991, −3.70093089349896061706983027467, −3.45992144781733985221268658583, −2.79639644855330517507098304064, −2.15293534837809612550077905006, −1.96819885237805176422041671183, −1.24271455357801539190571875522, 0,
1.24271455357801539190571875522, 1.96819885237805176422041671183, 2.15293534837809612550077905006, 2.79639644855330517507098304064, 3.45992144781733985221268658583, 3.70093089349896061706983027467, 4.34563034217913995952762795991, 4.70329769108401126404570380961, 5.40844333678785302124910491259, 5.79202265149903668260362805735, 6.23404778308086977014137472094, 6.63216653749155622423965381578, 7.03189212108301260064437659817, 7.30661797541290821455696261649
