| L(s) = 1 | − 2-s − 5-s − 7-s + 8-s + 10-s + 2·11-s − 13-s + 14-s − 16-s − 2·19-s − 2·22-s + 23-s + 26-s + 35-s − 4·37-s + 2·38-s − 40-s − 41-s − 46-s + 47-s + 49-s − 2·53-s − 2·55-s − 56-s − 59-s + 64-s + 65-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s − 7-s + 8-s + 10-s + 2·11-s − 13-s + 14-s − 16-s − 2·19-s − 2·22-s + 23-s + 26-s + 35-s − 4·37-s + 2·38-s − 40-s − 41-s − 46-s + 47-s + 49-s − 2·53-s − 2·55-s − 56-s − 59-s + 64-s + 65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1677165560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1677165560\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.250648848181753343840710991336, −8.677963390654611546023830711172, −8.422186182482200567766248301543, −8.122059813772090837011813955465, −7.40954608538076134563481673061, −7.15489562445487458480781604003, −6.94895603657755259193624640769, −6.54809510236037711097521435193, −6.35755515385569860338094178747, −5.59159254068738686641822853341, −5.19779706484269263561797165010, −4.64139054334581828293505593926, −4.27755478882460657635391282139, −3.99523513629948252906578176199, −3.50679612418110058989646829639, −3.18867958611158477146406243618, −2.47154083436437042987522116496, −1.59632432546985261808095667094, −1.58439722650982509814668325634, −0.29806711929197286982440614761,
0.29806711929197286982440614761, 1.58439722650982509814668325634, 1.59632432546985261808095667094, 2.47154083436437042987522116496, 3.18867958611158477146406243618, 3.50679612418110058989646829639, 3.99523513629948252906578176199, 4.27755478882460657635391282139, 4.64139054334581828293505593926, 5.19779706484269263561797165010, 5.59159254068738686641822853341, 6.35755515385569860338094178747, 6.54809510236037711097521435193, 6.94895603657755259193624640769, 7.15489562445487458480781604003, 7.40954608538076134563481673061, 8.122059813772090837011813955465, 8.422186182482200567766248301543, 8.677963390654611546023830711172, 9.250648848181753343840710991336
