| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 5·5-s + 24·6-s + 32·8-s + 27·9-s + 20·10-s + 67·11-s + 72·12-s + 41·13-s + 30·15-s + 80·16-s − 92·17-s + 108·18-s + 43·19-s + 60·20-s + 268·22-s + 148·23-s + 192·24-s + 105·25-s + 164·26-s + 108·27-s + 77·29-s + 120·30-s − 520·31-s + 192·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s + 1.41·8-s + 9-s + 0.632·10-s + 1.83·11-s + 1.73·12-s + 0.874·13-s + 0.516·15-s + 5/4·16-s − 1.31·17-s + 1.41·18-s + 0.519·19-s + 0.670·20-s + 2.59·22-s + 1.34·23-s + 1.63·24-s + 0.839·25-s + 1.23·26-s + 0.769·27-s + 0.493·29-s + 0.730·30-s − 3.01·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.40811989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.40811989\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - p T - 16 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 67 T + 3448 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 41 T + 4478 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 92 T + 386 p T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 43 T + 11154 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 148 T + 24430 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 520 T + 125837 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 426 T + 171106 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 576 T + 242170 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 243 T + 309490 T^{2} + 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 200614 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 224 T + 461126 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 687 T + 678832 T^{2} - 687 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 472 T + 637018 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 921 T + 578188 T^{2} + 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 526 T + 1033727 T^{2} + 526 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 774 T + 966562 T^{2} - 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1953 T + 2366992 T^{2} - 1953 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.61928937766189516700473602515, −11.26210790227110590822649362582, −10.70072960797156192377753580874, −10.41077603785074395341135845991, −9.376366870658148044438146225460, −9.254354090169290807724536862266, −8.777837729575214754704983747068, −8.396430600301944052098955286730, −7.36898269588910679389757255290, −7.12045332732493184708807202723, −6.57498440464429209410729684394, −6.28089311095927773039040956992, −5.35799986281677978614115303868, −4.99337486459294989881633988683, −4.09853458961683236273498185345, −3.80153131226249017107293553164, −3.29253773680833650107632045981, −2.54143118521753646353758576358, −1.74232261310625663869333750966, −1.22960965069211260674322243655,
1.22960965069211260674322243655, 1.74232261310625663869333750966, 2.54143118521753646353758576358, 3.29253773680833650107632045981, 3.80153131226249017107293553164, 4.09853458961683236273498185345, 4.99337486459294989881633988683, 5.35799986281677978614115303868, 6.28089311095927773039040956992, 6.57498440464429209410729684394, 7.12045332732493184708807202723, 7.36898269588910679389757255290, 8.396430600301944052098955286730, 8.777837729575214754704983747068, 9.254354090169290807724536862266, 9.376366870658148044438146225460, 10.41077603785074395341135845991, 10.70072960797156192377753580874, 11.26210790227110590822649362582, 11.61928937766189516700473602515
