| L(s) = 1 | − 8·5-s + 7-s − 2·9-s + 38·25-s + 8·31-s − 8·35-s + 16·43-s + 16·45-s − 8·47-s + 49-s + 8·61-s − 2·63-s − 24·67-s − 5·81-s + 24·101-s − 24·103-s − 24·107-s + 12·113-s − 22·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 0.377·7-s − 2/3·9-s + 38/5·25-s + 1.43·31-s − 1.35·35-s + 2.43·43-s + 2.38·45-s − 1.16·47-s + 1/7·49-s + 1.02·61-s − 0.251·63-s − 2.93·67-s − 5/9·81-s + 2.38·101-s − 2.36·103-s − 2.32·107-s + 1.12·113-s − 2·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.928329697232844766585907221767, −8.188057214509100026281306554311, −7.983821232000002036742511485775, −7.72782003802833126145569309094, −7.14438434722230698367292292955, −6.78907197794275632795066073083, −6.02020417555534802053969461502, −5.24725702689372953190086654330, −4.50185552371175935345657331179, −4.34927860692831690306580619795, −3.81394948833072648510784180802, −3.14137792992390816749783057891, −2.73255899043208384705811500747, −0.980550638938267372961223820619, 0,
0.980550638938267372961223820619, 2.73255899043208384705811500747, 3.14137792992390816749783057891, 3.81394948833072648510784180802, 4.34927860692831690306580619795, 4.50185552371175935345657331179, 5.24725702689372953190086654330, 6.02020417555534802053969461502, 6.78907197794275632795066073083, 7.14438434722230698367292292955, 7.72782003802833126145569309094, 7.983821232000002036742511485775, 8.188057214509100026281306554311, 8.928329697232844766585907221767
