L(s) = 1 | + 8·7-s − 16·13-s − 32·19-s + 32·25-s − 88·31-s + 68·37-s − 80·43-s − 50·49-s − 100·61-s + 16·67-s − 32·73-s + 152·79-s − 128·91-s + 352·97-s + 56·103-s − 46·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + ⋯ |
L(s) = 1 | + 8/7·7-s − 1.23·13-s − 1.68·19-s + 1.27·25-s − 2.83·31-s + 1.83·37-s − 1.86·43-s − 1.02·49-s − 1.63·61-s + 0.238·67-s − 0.438·73-s + 1.92·79-s − 1.40·91-s + 3.62·97-s + 0.543·103-s − 0.380·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.650347501\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.650347501\) |
\(L(2)\) |
|
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 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 32 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 770 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1664 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1184 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4160 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7490 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15680 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 176 T + p^{2} T^{2} )^{2} \) |
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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
−10.87548642493860639251882709288, −10.38657295919216017914004676548, −9.904309879826644061867577309318, −9.303229594422882131084680659752, −9.017074122597167586942537555400, −8.585294933962246197013940123430, −7.920249674073304813048618304025, −7.75767836017893870097385507627, −7.27921972016935093144738772096, −6.48972727011012459093414816766, −6.46502233235709708214728186314, −5.50745688625015102449690600891, −5.13258638695520631463292853335, −4.63450192156492300358272123092, −4.32307144306747344205210136404, −3.48735849955437090272040029836, −2.87543926117971228935150249411, −1.93885643398718639691324804647, −1.80442980206657036402054405532, −0.47264380370924817835984330791,
0.47264380370924817835984330791, 1.80442980206657036402054405532, 1.93885643398718639691324804647, 2.87543926117971228935150249411, 3.48735849955437090272040029836, 4.32307144306747344205210136404, 4.63450192156492300358272123092, 5.13258638695520631463292853335, 5.50745688625015102449690600891, 6.46502233235709708214728186314, 6.48972727011012459093414816766, 7.27921972016935093144738772096, 7.75767836017893870097385507627, 7.920249674073304813048618304025, 8.585294933962246197013940123430, 9.017074122597167586942537555400, 9.303229594422882131084680659752, 9.904309879826644061867577309318, 10.38657295919216017914004676548, 10.87548642493860639251882709288