| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s − 4·8-s + 8·10-s + 2·11-s + 8·13-s + 5·16-s − 4·17-s + 4·19-s − 12·20-s − 4·22-s + 4·23-s + 2·25-s − 16·26-s − 8·29-s + 4·31-s − 6·32-s + 8·34-s − 4·37-s − 8·38-s + 16·40-s − 12·41-s − 4·43-s + 6·44-s − 8·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s − 1.41·8-s + 2.52·10-s + 0.603·11-s + 2.21·13-s + 5/4·16-s − 0.970·17-s + 0.917·19-s − 2.68·20-s − 0.852·22-s + 0.834·23-s + 2/5·25-s − 3.13·26-s − 1.48·29-s + 0.718·31-s − 1.06·32-s + 1.37·34-s − 0.657·37-s − 1.29·38-s + 2.52·40-s − 1.87·41-s − 0.609·43-s + 0.904·44-s − 1.17·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 136 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−7.46097777734601280385088201311, −7.34631135086647411826343601970, −6.84398101536375019088910068645, −6.82281213986347351775868191746, −6.16149469322688461261825954011, −6.14100495040915870233285696474, −5.41473216242024660065866511989, −5.32633217586220673048508144680, −4.50578519622270047206322049979, −4.35313128952855843786545992903, −3.77095578713166586613405112466, −3.60037246306004207729214130025, −3.17455506569084298960814445469, −3.09015424304817645981087603647, −1.96894898700370322778528809264, −1.96617408269351178127910299745, −1.09552030927412792905760582792, −1.08946590599935230753533481686, 0, 0,
1.08946590599935230753533481686, 1.09552030927412792905760582792, 1.96617408269351178127910299745, 1.96894898700370322778528809264, 3.09015424304817645981087603647, 3.17455506569084298960814445469, 3.60037246306004207729214130025, 3.77095578713166586613405112466, 4.35313128952855843786545992903, 4.50578519622270047206322049979, 5.32633217586220673048508144680, 5.41473216242024660065866511989, 6.14100495040915870233285696474, 6.16149469322688461261825954011, 6.82281213986347351775868191746, 6.84398101536375019088910068645, 7.34631135086647411826343601970, 7.46097777734601280385088201311
