| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 2·9-s + 8·11-s − 2·13-s + 4·15-s − 4·17-s − 2·19-s − 4·21-s − 4·23-s − 2·25-s + 6·27-s − 12·29-s + 16·33-s − 4·35-s − 4·37-s − 4·39-s − 4·41-s + 4·45-s + 8·47-s + 3·49-s − 8·51-s + 16·55-s − 4·57-s + 14·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 2/3·9-s + 2.41·11-s − 0.554·13-s + 1.03·15-s − 0.970·17-s − 0.458·19-s − 0.872·21-s − 0.834·23-s − 2/5·25-s + 1.15·27-s − 2.22·29-s + 2.78·33-s − 0.676·35-s − 0.657·37-s − 0.640·39-s − 0.624·41-s + 0.596·45-s + 1.16·47-s + 3/7·49-s − 1.12·51-s + 2.15·55-s − 0.529·57-s + 1.82·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.993080077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.993080077\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.260282345635217047685708769929, −9.179135136203010067356555842429, −8.807309861476097020478603952235, −8.602534359892997022984243825510, −7.81690195742428478450313270554, −7.63756827093698100445490159245, −6.89522059839765788203350770256, −6.75896148171687209993918813911, −6.37359018056753932294079625988, −6.07804449452129868857383712779, −5.39320987343294281344906593083, −5.08974866762411481682241647280, −4.13678475661070323248488290034, −4.05118870976998466461888311081, −3.63959842515486552117290584445, −3.18586398480422168867629946601, −2.21308119235759693907688349249, −2.19455741570117945582660795581, −1.66695557977721702328940537053, −0.67335939940414930241452650588,
0.67335939940414930241452650588, 1.66695557977721702328940537053, 2.19455741570117945582660795581, 2.21308119235759693907688349249, 3.18586398480422168867629946601, 3.63959842515486552117290584445, 4.05118870976998466461888311081, 4.13678475661070323248488290034, 5.08974866762411481682241647280, 5.39320987343294281344906593083, 6.07804449452129868857383712779, 6.37359018056753932294079625988, 6.75896148171687209993918813911, 6.89522059839765788203350770256, 7.63756827093698100445490159245, 7.81690195742428478450313270554, 8.602534359892997022984243825510, 8.807309861476097020478603952235, 9.179135136203010067356555842429, 9.260282345635217047685708769929
