| L(s) = 1 | − 3-s + 4-s + 4·7-s − 2·9-s − 12-s + 16-s + 2·19-s − 4·21-s + 25-s + 5·27-s + 4·28-s − 2·36-s − 6·41-s + 10·43-s − 48-s + 7·49-s − 12·53-s − 2·57-s + 6·59-s + 10·61-s − 8·63-s + 64-s + 6·71-s − 2·73-s − 75-s + 2·76-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1.51·7-s − 2/3·9-s − 0.288·12-s + 1/4·16-s + 0.458·19-s − 0.872·21-s + 1/5·25-s + 0.962·27-s + 0.755·28-s − 1/3·36-s − 0.937·41-s + 1.52·43-s − 0.144·48-s + 49-s − 1.64·53-s − 0.264·57-s + 0.781·59-s + 1.28·61-s − 1.00·63-s + 1/8·64-s + 0.712·71-s − 0.234·73-s − 0.115·75-s + 0.229·76-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913580043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913580043\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.565929136878549365807251193340, −8.376100285709741390012149447232, −7.79857314240323647012025181375, −7.48684855355639228898209155521, −6.85950664583279958465505957183, −6.44250787353207025252245590662, −5.80201472204536922908320729557, −5.47986445078808467825167546354, −4.91688825351409857270451782239, −4.59787972103688108306579185017, −3.78844256551048566972058875799, −3.12563597140545438110769135332, −2.40069242771985848465965978158, −1.75223148945196589214164913087, −0.856792700842364074085854575706,
0.856792700842364074085854575706, 1.75223148945196589214164913087, 2.40069242771985848465965978158, 3.12563597140545438110769135332, 3.78844256551048566972058875799, 4.59787972103688108306579185017, 4.91688825351409857270451782239, 5.47986445078808467825167546354, 5.80201472204536922908320729557, 6.44250787353207025252245590662, 6.85950664583279958465505957183, 7.48684855355639228898209155521, 7.79857314240323647012025181375, 8.376100285709741390012149447232, 8.565929136878549365807251193340
