| L(s) = 1 | − 2·4-s + 6·7-s + 6·13-s + 2·19-s − 12·28-s + 4·31-s + 18·37-s + 12·43-s + 13·49-s − 12·52-s − 26·61-s + 8·64-s − 6·67-s + 18·73-s − 4·76-s − 10·79-s + 36·91-s + 6·97-s + 6·103-s − 16·109-s − 4·121-s − 8·124-s + 127-s + 131-s + 12·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 2.26·7-s + 1.66·13-s + 0.458·19-s − 2.26·28-s + 0.718·31-s + 2.95·37-s + 1.82·43-s + 13/7·49-s − 1.66·52-s − 3.32·61-s + 64-s − 0.733·67-s + 2.10·73-s − 0.458·76-s − 1.12·79-s + 3.77·91-s + 0.609·97-s + 0.591·103-s − 1.53·109-s − 0.363·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 1.04·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.324289234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.324289234\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
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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
−10.96359787634588376615553466527, −10.42363591948263018730690249962, −9.675549953362133828139338771814, −9.354568816709515276368823300071, −8.918690775052199079847952077310, −8.602976094239182261871599682404, −7.973478575193843688867603196486, −7.88733753718221590586782687106, −7.56086427063420874066164790829, −6.67129509723357145602197202319, −6.00525604869664938262318316201, −5.88126166141498480956011352016, −5.02285520677113331935856891020, −4.78750080225639109817325650821, −4.19659819879050368867774418745, −4.06388630062889194738217853710, −3.07835625551688383837876718067, −2.35055555254869265430025024889, −1.41968084240141269635968831838, −0.991594080094343570012066853212,
0.991594080094343570012066853212, 1.41968084240141269635968831838, 2.35055555254869265430025024889, 3.07835625551688383837876718067, 4.06388630062889194738217853710, 4.19659819879050368867774418745, 4.78750080225639109817325650821, 5.02285520677113331935856891020, 5.88126166141498480956011352016, 6.00525604869664938262318316201, 6.67129509723357145602197202319, 7.56086427063420874066164790829, 7.88733753718221590586782687106, 7.973478575193843688867603196486, 8.602976094239182261871599682404, 8.918690775052199079847952077310, 9.354568816709515276368823300071, 9.675549953362133828139338771814, 10.42363591948263018730690249962, 10.96359787634588376615553466527
