| L(s) = 1 | + 4-s − 7-s − 5·9-s − 4·11-s − 2·13-s + 16-s + 10·17-s − 6·25-s − 28-s − 6·29-s − 5·36-s − 6·37-s − 4·44-s − 4·47-s − 6·49-s − 2·52-s − 10·53-s − 8·59-s + 5·63-s + 64-s + 10·68-s + 4·77-s + 16·81-s + 4·89-s + 2·91-s + 10·97-s + 20·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.377·7-s − 5/3·9-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 2.42·17-s − 6/5·25-s − 0.188·28-s − 1.11·29-s − 5/6·36-s − 0.986·37-s − 0.603·44-s − 0.583·47-s − 6/7·49-s − 0.277·52-s − 1.37·53-s − 1.04·59-s + 0.629·63-s + 1/8·64-s + 1.21·68-s + 0.455·77-s + 16/9·81-s + 0.423·89-s + 0.209·91-s + 1.01·97-s + 2.01·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 117 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.423282324701479242666600606769, −9.206480350268086051667131457369, −8.222497627822697132021546641534, −7.953984232130324613375788150716, −7.68922980749156452332992011692, −7.04512790380869864485879441480, −6.14973405640971963464029046625, −5.89577345968739888220551014333, −5.28062086983382278817636428768, −5.02316167718045361032079495467, −3.70196667116925686304663289954, −3.19845435073716837061198501807, −2.75007023879120036211234519378, −1.78151251755801704712845611529, 0,
1.78151251755801704712845611529, 2.75007023879120036211234519378, 3.19845435073716837061198501807, 3.70196667116925686304663289954, 5.02316167718045361032079495467, 5.28062086983382278817636428768, 5.89577345968739888220551014333, 6.14973405640971963464029046625, 7.04512790380869864485879441480, 7.68922980749156452332992011692, 7.953984232130324613375788150716, 8.222497627822697132021546641534, 9.206480350268086051667131457369, 9.423282324701479242666600606769
