L(s) = 1 | + 2·5-s + 5·9-s + 12·11-s − 14·19-s − 25-s + 10·29-s − 10·31-s + 10·45-s + 14·49-s + 24·55-s + 10·59-s − 6·61-s + 30·71-s + 16·79-s + 16·81-s + 2·89-s − 28·95-s + 60·99-s − 24·101-s − 18·109-s + 86·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 5/3·9-s + 3.61·11-s − 3.21·19-s − 1/5·25-s + 1.85·29-s − 1.79·31-s + 1.49·45-s + 2·49-s + 3.23·55-s + 1.30·59-s − 0.768·61-s + 3.56·71-s + 1.80·79-s + 16/9·81-s + 0.211·89-s − 2.87·95-s + 6.03·99-s − 2.38·101-s − 1.72·109-s + 7.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.206447095\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.206447095\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
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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
−9.570690020225704983158418162991, −9.384327207690774629814647312096, −9.192574522820162040018660793224, −8.675694448516752966734918089378, −8.360093713182197486207385378734, −7.83556236856731453711704712607, −6.95653800860170315055377824051, −6.82358228627235479371990021853, −6.60626104899568260649524988969, −6.33327852515320439469370608729, −5.82435807838363348966214077516, −5.18299696410860250584645653824, −4.35080978276390976713497611681, −4.32691959818692482434743468213, −3.78634961933510770782872245528, −3.67075523301643659964076752895, −2.20085766247182498621485734973, −2.15024537705980424116003593216, −1.43048333089379691180009463211, −0.961201824584201255857880365572,
0.961201824584201255857880365572, 1.43048333089379691180009463211, 2.15024537705980424116003593216, 2.20085766247182498621485734973, 3.67075523301643659964076752895, 3.78634961933510770782872245528, 4.32691959818692482434743468213, 4.35080978276390976713497611681, 5.18299696410860250584645653824, 5.82435807838363348966214077516, 6.33327852515320439469370608729, 6.60626104899568260649524988969, 6.82358228627235479371990021853, 6.95653800860170315055377824051, 7.83556236856731453711704712607, 8.360093713182197486207385378734, 8.675694448516752966734918089378, 9.192574522820162040018660793224, 9.384327207690774629814647312096, 9.570690020225704983158418162991