| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s − 6·17-s − 2·23-s + 25-s − 4·27-s + 10·29-s − 3·36-s − 12·39-s − 2·43-s − 2·48-s + 12·51-s − 6·52-s + 28·53-s + 6·61-s − 64-s + 6·68-s + 4·69-s − 2·75-s + 5·81-s − 20·87-s + 2·92-s − 100-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s − 1.45·17-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s − 1/2·36-s − 1.92·39-s − 0.304·43-s − 0.288·48-s + 1.68·51-s − 0.832·52-s + 3.84·53-s + 0.768·61-s − 1/8·64-s + 0.727·68-s + 0.481·69-s − 0.230·75-s + 5/9·81-s − 2.14·87-s + 0.208·92-s − 0.0999·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.814912666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.814912666\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.657125988312815435372014826257, −8.582577335381216807710173931707, −7.77840636651054930316797343792, −7.72782540541938451114662998913, −6.89803646769211186237488301309, −6.88492420145800632857061378353, −6.37673355583598373350023667169, −5.96849787516930691220951283719, −5.95788415156388779443713366608, −5.12365964625832896332202733507, −5.06168442479569709608310555560, −4.55996690636076979084861311869, −4.02355196865025184086112996245, −3.91434705374179076335939784827, −3.38779819560221011376225504792, −2.66683924418493550889596527714, −2.19555752122016092262001852524, −1.59272909353369140486450796570, −0.813828937525400171009416244286, −0.62448721802453955512396882995,
0.62448721802453955512396882995, 0.813828937525400171009416244286, 1.59272909353369140486450796570, 2.19555752122016092262001852524, 2.66683924418493550889596527714, 3.38779819560221011376225504792, 3.91434705374179076335939784827, 4.02355196865025184086112996245, 4.55996690636076979084861311869, 5.06168442479569709608310555560, 5.12365964625832896332202733507, 5.95788415156388779443713366608, 5.96849787516930691220951283719, 6.37673355583598373350023667169, 6.88492420145800632857061378353, 6.89803646769211186237488301309, 7.72782540541938451114662998913, 7.77840636651054930316797343792, 8.582577335381216807710173931707, 8.657125988312815435372014826257
