| L(s) = 1 | − 2·3-s − 4·4-s + 3·9-s + 8·11-s + 8·12-s + 4·13-s + 12·16-s + 10·17-s + 8·19-s + 4·23-s + 2·25-s − 4·27-s + 8·31-s − 16·33-s − 12·36-s − 2·37-s − 8·39-s − 2·41-s + 6·43-s − 32·44-s − 10·47-s − 24·48-s − 20·51-s − 16·52-s + 8·53-s − 16·57-s − 32·64-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s + 9-s + 2.41·11-s + 2.30·12-s + 1.10·13-s + 3·16-s + 2.42·17-s + 1.83·19-s + 0.834·23-s + 2/5·25-s − 0.769·27-s + 1.43·31-s − 2.78·33-s − 2·36-s − 0.328·37-s − 1.28·39-s − 0.312·41-s + 0.914·43-s − 4.82·44-s − 1.45·47-s − 3.46·48-s − 2.80·51-s − 2.21·52-s + 1.09·53-s − 2.11·57-s − 4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36324729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.609763380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.609763380\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 75 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 119 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 179 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.144811068668566753616389937071, −8.079579061605474146839765528802, −7.48305768886158733058649345782, −7.29711610750205640907711380364, −6.62734796508184780457922494309, −6.48388541294054542856372076272, −5.96092619600802254633363645041, −5.74027157001462864198102729449, −5.32053519195029434308009093314, −5.06839050773537789753986645949, −4.72325506193205236395564056509, −4.31944662628808142448542673249, −3.68081390440697351745747460980, −3.65467132405764808769739900004, −3.42129821682627751034518720275, −2.82396952460461692817302116283, −1.53309842138885468730798388190, −1.24591496377777788773578214664, −0.884977588474042459623895640318, −0.75815891950555547584392872027,
0.75815891950555547584392872027, 0.884977588474042459623895640318, 1.24591496377777788773578214664, 1.53309842138885468730798388190, 2.82396952460461692817302116283, 3.42129821682627751034518720275, 3.65467132405764808769739900004, 3.68081390440697351745747460980, 4.31944662628808142448542673249, 4.72325506193205236395564056509, 5.06839050773537789753986645949, 5.32053519195029434308009093314, 5.74027157001462864198102729449, 5.96092619600802254633363645041, 6.48388541294054542856372076272, 6.62734796508184780457922494309, 7.29711610750205640907711380364, 7.48305768886158733058649345782, 8.079579061605474146839765528802, 8.144811068668566753616389937071
