| L(s) = 1 | − 4-s − 4·7-s − 6·9-s + 12·11-s + 16-s + 4·28-s + 6·36-s − 2·37-s − 14·41-s − 12·44-s − 8·47-s − 2·49-s − 6·53-s + 24·63-s − 64-s − 28·67-s + 16·71-s − 18·73-s − 48·77-s + 27·81-s − 24·83-s − 72·99-s − 2·101-s − 20·107-s − 4·112-s + 86·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.51·7-s − 2·9-s + 3.61·11-s + 1/4·16-s + 0.755·28-s + 36-s − 0.328·37-s − 2.18·41-s − 1.80·44-s − 1.16·47-s − 2/7·49-s − 0.824·53-s + 3.02·63-s − 1/8·64-s − 3.42·67-s + 1.89·71-s − 2.10·73-s − 5.47·77-s + 3·81-s − 2.63·83-s − 7.23·99-s − 0.199·101-s − 1.93·107-s − 0.377·112-s + 7.81·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6140772552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6140772552\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.492927460321953177425962573893, −9.059723074413026063399806388984, −8.742642873067948824036569341240, −8.410135851377883879307673714386, −8.193902532341778650203784334616, −7.25928666614700298441029520591, −6.94614472531960212928529258931, −6.53147526981145297466179210259, −6.33915090559610223091714956230, −5.84472225680866520765131160147, −5.73842098004048501468939457138, −4.81341076677979470904845161691, −4.55641307840245125759115380236, −3.81268458508326018939652207343, −3.61040123804195863526863655823, −3.05353724291567430385296592837, −2.98094459451298021843431971127, −1.71238437737382107251882819864, −1.41698754396609981839457845652, −0.29875353737744996307474456198,
0.29875353737744996307474456198, 1.41698754396609981839457845652, 1.71238437737382107251882819864, 2.98094459451298021843431971127, 3.05353724291567430385296592837, 3.61040123804195863526863655823, 3.81268458508326018939652207343, 4.55641307840245125759115380236, 4.81341076677979470904845161691, 5.73842098004048501468939457138, 5.84472225680866520765131160147, 6.33915090559610223091714956230, 6.53147526981145297466179210259, 6.94614472531960212928529258931, 7.25928666614700298441029520591, 8.193902532341778650203784334616, 8.410135851377883879307673714386, 8.742642873067948824036569341240, 9.059723074413026063399806388984, 9.492927460321953177425962573893
