| L(s) = 1 | + 10·3-s + 8·5-s + 27·9-s + 40·11-s − 24·13-s + 80·15-s + 58·17-s − 26·19-s + 64·23-s + 125·25-s − 190·27-s − 124·29-s − 252·31-s + 400·33-s − 26·37-s − 240·39-s + 12·41-s + 832·43-s + 216·45-s + 396·47-s + 580·51-s + 450·53-s + 320·55-s − 260·57-s − 274·59-s + 576·61-s − 192·65-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.715·5-s + 9-s + 1.09·11-s − 0.512·13-s + 1.37·15-s + 0.827·17-s − 0.313·19-s + 0.580·23-s + 25-s − 1.35·27-s − 0.794·29-s − 1.46·31-s + 2.11·33-s − 0.115·37-s − 0.985·39-s + 0.0457·41-s + 2.95·43-s + 0.715·45-s + 1.22·47-s + 1.59·51-s + 1.16·53-s + 0.784·55-s − 0.604·57-s − 0.604·59-s + 1.20·61-s − 0.366·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.681446318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.681446318\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 10 T + 73 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T - 61 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 40 T + 269 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 58 T - 1549 T^{2} - 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T - 6183 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 64 T - 8071 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 252 T + 33713 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T - 49977 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 396 T + 52993 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 450 T + 53623 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 274 T - 130303 T^{2} + 274 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 576 T + 104795 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 476 T - 74187 T^{2} - 476 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 448 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 158 T - 364053 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 936 T + 383057 T^{2} - 936 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 530 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 390 T - 552869 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 214 T + p^{3} T^{2} )^{2} \) |
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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
−12.43755530188440614928341334468, −11.98470222142444301110988160710, −11.03710599351132318421050379612, −10.96315528129379905217966183099, −10.10633275032607655611204346792, −9.390649417109911477062086456869, −9.385430617173997507533918460550, −8.737257793297386489937285922803, −8.625963226495651196041468102828, −7.57970861253101630942520882596, −7.49417516315306627491714745874, −6.79487698336805675792199409554, −5.79841274672679197589537583175, −5.64568051608364021810242730611, −4.58451412592688943143243306851, −3.72429166374827658842227593687, −3.39983951253510365262967404795, −2.43235131797311039427752703867, −2.15501281059897491347571402892, −0.976235712202319232569156427841,
0.976235712202319232569156427841, 2.15501281059897491347571402892, 2.43235131797311039427752703867, 3.39983951253510365262967404795, 3.72429166374827658842227593687, 4.58451412592688943143243306851, 5.64568051608364021810242730611, 5.79841274672679197589537583175, 6.79487698336805675792199409554, 7.49417516315306627491714745874, 7.57970861253101630942520882596, 8.625963226495651196041468102828, 8.737257793297386489937285922803, 9.385430617173997507533918460550, 9.390649417109911477062086456869, 10.10633275032607655611204346792, 10.96315528129379905217966183099, 11.03710599351132318421050379612, 11.98470222142444301110988160710, 12.43755530188440614928341334468
