| L(s) = 1 | + 2·3-s − 4·4-s + 6·5-s − 3·9-s − 11-s − 8·12-s + 12·15-s + 12·16-s − 24·20-s + 6·23-s + 17·25-s − 14·27-s + 10·31-s − 2·33-s + 12·36-s + 22·37-s + 4·44-s − 18·45-s + 24·48-s + 49-s − 12·53-s − 6·55-s − 18·59-s − 48·60-s − 32·64-s + 10·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2·4-s + 2.68·5-s − 9-s − 0.301·11-s − 2.30·12-s + 3.09·15-s + 3·16-s − 5.36·20-s + 1.25·23-s + 17/5·25-s − 2.69·27-s + 1.79·31-s − 0.348·33-s + 2·36-s + 3.61·37-s + 0.603·44-s − 2.68·45-s + 3.46·48-s + 1/7·49-s − 1.64·53-s − 0.809·55-s − 2.34·59-s − 6.19·60-s − 4·64-s + 1.22·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933947068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933947068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.594904136117515823873522897188, −9.487459835858535121288790482109, −9.121939609810073781754575230776, −8.508833833811404945218630536704, −8.085327774959512351361821751845, −7.79543673470654027074280114997, −6.46084267575588440942544784885, −5.89237731397243587720148634695, −5.81865396854508245636334597208, −4.95929836259470425310107953583, −4.69940224529243356978622440045, −3.61592205748787980286928494613, −2.75685822824512438972431635156, −2.52489627324288051568453078479, −1.19698223664384976616389110079,
1.19698223664384976616389110079, 2.52489627324288051568453078479, 2.75685822824512438972431635156, 3.61592205748787980286928494613, 4.69940224529243356978622440045, 4.95929836259470425310107953583, 5.81865396854508245636334597208, 5.89237731397243587720148634695, 6.46084267575588440942544784885, 7.79543673470654027074280114997, 8.085327774959512351361821751845, 8.508833833811404945218630536704, 9.121939609810073781754575230776, 9.487459835858535121288790482109, 9.594904136117515823873522897188
