| L(s) = 1 | + 6·3-s + 27·9-s − 96·11-s − 84·17-s + 184·19-s + 146·25-s + 108·27-s − 576·33-s + 12·41-s + 184·43-s − 290·49-s − 504·51-s + 1.10e3·57-s − 1.03e3·59-s − 1.04e3·67-s + 860·73-s + 876·75-s + 405·81-s − 864·83-s − 1.26e3·89-s + 1.72e3·97-s − 2.59e3·99-s − 2.76e3·107-s + 4.21e3·113-s + 4.25e3·121-s + 72·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.63·11-s − 1.19·17-s + 2.22·19-s + 1.16·25-s + 0.769·27-s − 3.03·33-s + 0.0457·41-s + 0.652·43-s − 0.845·49-s − 1.38·51-s + 2.56·57-s − 2.27·59-s − 1.91·67-s + 1.37·73-s + 1.34·75-s + 5/9·81-s − 1.14·83-s − 1.50·89-s + 1.80·97-s − 2.63·99-s − 2.49·107-s + 3.50·113-s + 3.19·121-s + 0.0527·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.560521258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.560521258\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 146 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 290 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1942 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22750 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 48382 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 40178 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61706 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 206062 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50254 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 516 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 325658 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 524 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 274178 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 430 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 392398 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 862 T + p^{3} T^{2} )^{2} \) |
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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
−9.923197924784733136958132195661, −9.845531640691913664543343828852, −9.095147622484556597005338696068, −9.020618563979615182111949787431, −8.320148017121371474615024319677, −8.034854300454489148450223921630, −7.50056771707639292334891622618, −7.44145408819017808587841127649, −6.85812472647848799574257023941, −6.20276611259302507944279795599, −5.54558903591099491105075867400, −5.20796605339152220729150880329, −4.64775592970434749139599751605, −4.36757084190291984414072351430, −3.23031506611215807303950327115, −3.13601652017886740431793069421, −2.66409682345466495325816226579, −2.07805033807554460098575897583, −1.30583637796795972179117655634, −0.39940529911352392591634383942,
0.39940529911352392591634383942, 1.30583637796795972179117655634, 2.07805033807554460098575897583, 2.66409682345466495325816226579, 3.13601652017886740431793069421, 3.23031506611215807303950327115, 4.36757084190291984414072351430, 4.64775592970434749139599751605, 5.20796605339152220729150880329, 5.54558903591099491105075867400, 6.20276611259302507944279795599, 6.85812472647848799574257023941, 7.44145408819017808587841127649, 7.50056771707639292334891622618, 8.034854300454489148450223921630, 8.320148017121371474615024319677, 9.020618563979615182111949787431, 9.095147622484556597005338696068, 9.845531640691913664543343828852, 9.923197924784733136958132195661
