| L(s) = 1 | + 2·5-s − 3·9-s − 3·11-s + 9·23-s − 3·25-s + 31-s + 6·37-s − 6·45-s − 13·47-s − 9·49-s − 2·53-s − 6·55-s + 5·59-s + 14·67-s − 16·71-s + 9·81-s + 30·89-s − 15·97-s + 9·99-s + 103-s + 13·113-s + 18·115-s − 2·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s − 0.904·11-s + 1.87·23-s − 3/5·25-s + 0.179·31-s + 0.986·37-s − 0.894·45-s − 1.89·47-s − 9/7·49-s − 0.274·53-s − 0.809·55-s + 0.650·59-s + 1.71·67-s − 1.89·71-s + 81-s + 3.17·89-s − 1.52·97-s + 0.904·99-s + 0.0985·103-s + 1.22·113-s + 1.67·115-s − 0.181·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 69 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−7.44151126668681669524687792259, −7.00878039269013825282439491017, −6.47758496866639655111158863524, −6.10250463880629689655326743159, −5.84956439995286640234130923052, −5.21958689456120862847576080331, −4.92331167347023792234908667206, −4.70707363248878522957341794804, −3.72880604582316828812687327597, −3.35068255680052480913720138864, −2.76952003112965008143634669029, −2.44080595567376607762390601798, −1.79647520171410037571270294532, −1.02123098984004542739766011608, 0,
1.02123098984004542739766011608, 1.79647520171410037571270294532, 2.44080595567376607762390601798, 2.76952003112965008143634669029, 3.35068255680052480913720138864, 3.72880604582316828812687327597, 4.70707363248878522957341794804, 4.92331167347023792234908667206, 5.21958689456120862847576080331, 5.84956439995286640234130923052, 6.10250463880629689655326743159, 6.47758496866639655111158863524, 7.00878039269013825282439491017, 7.44151126668681669524687792259
