L(s) = 1 | − 968·7-s − 6.73e3·13-s − 1.14e4·19-s + 992·25-s − 7.95e4·31-s − 1.05e5·37-s − 7.60e3·43-s + 4.67e5·49-s − 2.65e4·61-s − 3.37e5·67-s + 4.72e5·73-s − 7.02e4·79-s + 6.52e6·91-s − 6.42e5·97-s + 3.98e6·103-s − 3.88e5·109-s + 1.74e6·121-s + 127-s + 131-s + 1.11e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 2.82·7-s − 3.06·13-s − 1.67·19-s + 0.0634·25-s − 2.67·31-s − 2.07·37-s − 0.0955·43-s + 3.97·49-s − 0.116·61-s − 1.12·67-s + 1.21·73-s − 0.142·79-s + 8.65·91-s − 0.704·97-s + 3.64·103-s − 0.300·109-s + 0.985·121-s + 4.72·133-s + 5.05·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.01743197464\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01743197464\) |
\(L(4)\) |
|
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 | $C_2^2$ | \( 1 - 992 T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 484 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1745714 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3368 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 48274976 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5744 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 284666690 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 327940544 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 39796 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 52526 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8128061984 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 3800 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 15661450658 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12666935680 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 21940729490 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13250 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 168968 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 26256724990 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 236144 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 35116 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 653760187346 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 977236742720 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 321424 T + p^{6} T^{2} )^{2} \) |
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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
−10.27876645180557098915012920921, −9.877492568020404259030051185343, −9.094250275509000795959074762102, −8.985396480364529280009836643966, −8.321870104776695269627931387523, −7.37436806806884325193501357667, −7.22028686230430019203986614233, −7.05743335762494203404048075634, −6.27210631400804706917432337395, −6.17520176130796224768196222728, −5.33529193464473251429447855219, −5.03548391886707498778162653087, −4.33939507987719952692668634255, −3.72432729673021651103889937211, −3.26504092334992315888035816923, −2.87513367723826408985859619115, −2.05765000041784171102140571965, −2.02985928224993188831161217903, −0.49579283441015451900999534308, −0.04823688285102974160036775537,
0.04823688285102974160036775537, 0.49579283441015451900999534308, 2.02985928224993188831161217903, 2.05765000041784171102140571965, 2.87513367723826408985859619115, 3.26504092334992315888035816923, 3.72432729673021651103889937211, 4.33939507987719952692668634255, 5.03548391886707498778162653087, 5.33529193464473251429447855219, 6.17520176130796224768196222728, 6.27210631400804706917432337395, 7.05743335762494203404048075634, 7.22028686230430019203986614233, 7.37436806806884325193501357667, 8.321870104776695269627931387523, 8.985396480364529280009836643966, 9.094250275509000795959074762102, 9.877492568020404259030051185343, 10.27876645180557098915012920921