| L(s) = 1 | − 3·3-s − 6·5-s + 10·7-s + 6·9-s − 33·13-s + 18·15-s − 6·17-s + 26·19-s − 30·21-s − 24·23-s + 25·25-s − 9·27-s + 54·29-s − 60·35-s + 99·39-s − 72·41-s − 25·43-s − 36·45-s − 42·47-s − 23·49-s + 18·51-s + 108·53-s − 78·57-s − 126·59-s − 43·61-s + 60·63-s + 198·65-s + ⋯ |
| L(s) = 1 | − 3-s − 6/5·5-s + 10/7·7-s + 2/3·9-s − 2.53·13-s + 6/5·15-s − 0.352·17-s + 1.36·19-s − 1.42·21-s − 1.04·23-s + 25-s − 1/3·27-s + 1.86·29-s − 1.71·35-s + 2.53·39-s − 1.75·41-s − 0.581·43-s − 4/5·45-s − 0.893·47-s − 0.469·49-s + 6/17·51-s + 2.03·53-s − 1.36·57-s − 2.13·59-s − 0.704·61-s + 0.952·63-s + 3.04·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02379281490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02379281490\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 26 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 33 T + 532 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T - 253 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 24 T + 47 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T + 1813 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1055 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 937 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 72 T + 3409 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 25 T - 1224 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 42 T - 445 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 108 T + 6697 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 126 T + 8773 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 43 T - 1872 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 99 T + 7756 T^{2} + 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 108 T + 8929 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T - 5208 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T + 6244 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 8029 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 228 T + 26737 T^{2} + 228 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.32665792727635543057324546143, −9.854031842611033756411292585747, −9.402694254736172943927003576163, −8.622889102468590100163978499284, −8.408258494524091025317623760409, −7.73774751553201633396306582281, −7.73657486242439799146546842057, −7.06158990459906161662017566222, −6.95128026036942722069777889816, −6.26513088224085724924564731362, −5.57259733977194787694948563650, −5.02324834279201588034082146053, −4.96793362982337263788817556352, −4.34712322953427252703238614140, −4.21938234099063058897154533427, −2.95630584492293215192943945633, −2.92004431565720049781975229487, −1.76791714782123571490302283353, −1.29637605717862797527193283291, −0.05610037965104912722668588883,
0.05610037965104912722668588883, 1.29637605717862797527193283291, 1.76791714782123571490302283353, 2.92004431565720049781975229487, 2.95630584492293215192943945633, 4.21938234099063058897154533427, 4.34712322953427252703238614140, 4.96793362982337263788817556352, 5.02324834279201588034082146053, 5.57259733977194787694948563650, 6.26513088224085724924564731362, 6.95128026036942722069777889816, 7.06158990459906161662017566222, 7.73657486242439799146546842057, 7.73774751553201633396306582281, 8.408258494524091025317623760409, 8.622889102468590100163978499284, 9.402694254736172943927003576163, 9.854031842611033756411292585747, 10.32665792727635543057324546143
