| L(s) = 1 | − 2·3-s + 4-s + 2·5-s + 3·9-s − 2·12-s − 4·15-s − 3·16-s + 2·20-s + 4·23-s + 3·25-s − 4·27-s − 4·31-s + 3·36-s − 8·37-s + 6·45-s − 12·47-s + 6·48-s + 6·49-s − 8·53-s − 4·59-s − 4·60-s − 7·64-s + 16·67-s − 8·69-s + 16·71-s − 6·75-s − 6·80-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.894·5-s + 9-s − 0.577·12-s − 1.03·15-s − 3/4·16-s + 0.447·20-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 0.718·31-s + 1/2·36-s − 1.31·37-s + 0.894·45-s − 1.75·47-s + 0.866·48-s + 6/7·49-s − 1.09·53-s − 0.520·59-s − 0.516·60-s − 7/8·64-s + 1.95·67-s − 0.963·69-s + 1.89·71-s − 0.692·75-s − 0.670·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.583278428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583278428\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.23815392841040708057024452919, −6.93369534054451136237238298192, −6.66997089871483937530094116207, −6.33441546971782071295643763649, −5.87511436039108526070458178589, −5.35744616282653305288898933553, −5.14595033301132747372216851028, −4.75081564435104570741968555343, −4.18646610451943193090708612222, −3.56281303427106806080518253523, −3.07807572238901504119232779846, −2.38889252462389521168499293084, −1.86247893551332268542445079595, −1.42566135957896737632755522849, −0.50211099024170605810363493885,
0.50211099024170605810363493885, 1.42566135957896737632755522849, 1.86247893551332268542445079595, 2.38889252462389521168499293084, 3.07807572238901504119232779846, 3.56281303427106806080518253523, 4.18646610451943193090708612222, 4.75081564435104570741968555343, 5.14595033301132747372216851028, 5.35744616282653305288898933553, 5.87511436039108526070458178589, 6.33441546971782071295643763649, 6.66997089871483937530094116207, 6.93369534054451136237238298192, 7.23815392841040708057024452919
