L(s) = 1 | + 4·2-s − 3-s + 8·4-s − 4·6-s + 8·8-s − 2·9-s − 11-s − 8·12-s − 4·16-s + 4·17-s − 8·18-s − 4·22-s − 8·24-s − 9·25-s + 5·27-s + 14·31-s − 32·32-s + 33-s + 16·34-s − 16·36-s + 6·37-s + 16·41-s − 8·44-s + 4·48-s − 10·49-s − 36·50-s − 4·51-s + ⋯ |
L(s) = 1 | + 2.82·2-s − 0.577·3-s + 4·4-s − 1.63·6-s + 2.82·8-s − 2/3·9-s − 0.301·11-s − 2.30·12-s − 16-s + 0.970·17-s − 1.88·18-s − 0.852·22-s − 1.63·24-s − 9/5·25-s + 0.962·27-s + 2.51·31-s − 5.65·32-s + 0.174·33-s + 2.74·34-s − 8/3·36-s + 0.986·37-s + 2.49·41-s − 1.20·44-s + 0.577·48-s − 1.42·49-s − 5.09·50-s − 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.963701228\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.963701228\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−11.81631087306955963420628700213, −11.14136051051457299622132768480, −10.76517194615392112496312976161, −9.701226498296212058900989539920, −9.366222803316740054530803267641, −8.228854358433494441874279371750, −7.78789607043645765017922373874, −6.66503987304844816686807720890, −6.23870064789214498217427813253, −5.66407660024727617127120629681, −5.41175071356617719099292784896, −4.44574490854809981150981932393, −4.19085115174357929506261869806, −3.09000916592887094247460325417, −2.63044898935838963010520851422,
2.63044898935838963010520851422, 3.09000916592887094247460325417, 4.19085115174357929506261869806, 4.44574490854809981150981932393, 5.41175071356617719099292784896, 5.66407660024727617127120629681, 6.23870064789214498217427813253, 6.66503987304844816686807720890, 7.78789607043645765017922373874, 8.228854358433494441874279371750, 9.366222803316740054530803267641, 9.701226498296212058900989539920, 10.76517194615392112496312976161, 11.14136051051457299622132768480, 11.81631087306955963420628700213