| L(s) = 1 | − 4.09e3·2-s + 8.38e6·4-s + 7.11e7·5-s − 2.91e11·10-s − 1.80e13·13-s − 7.03e13·16-s + 5.44e13·17-s + 5.96e14·20-s − 6.86e15·25-s + 7.37e16·26-s + 2.88e17·32-s − 2.23e17·34-s + 3.92e17·37-s + 2.64e18·41-s + 2.81e19·50-s − 1.51e20·52-s + 1.18e20·53-s − 1.30e21·61-s − 5.90e20·64-s − 1.28e21·65-s + 4.56e20·68-s + 7.01e21·73-s − 1.60e21·74-s − 5.00e21·80-s − 8.86e21·81-s − 1.08e22·82-s + 3.87e21·85-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.651·5-s − 0.921·10-s − 2.78·13-s − 16-s + 0.385·17-s + 0.651·20-s − 0.575·25-s + 3.94·26-s + 1.41·32-s − 0.545·34-s + 0.362·37-s + 0.751·41-s + 0.814·50-s − 2.78·52-s + 1.75·53-s − 3.85·61-s − 64-s − 1.81·65-s + 0.385·68-s + 2.61·73-s − 0.512·74-s − 0.651·80-s − 81-s − 1.06·82-s + 0.251·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(0.2989204162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2989204162\) |
| \(L(\frac{25}{2})\) |
|
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 | $C_2$ | \( 1 + p^{12} T + p^{23} T^{2} \) |
| 5 | $C_2$ | \( 1 - 71106796 T + p^{23} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7436301651582 T + p^{23} T^{2} )( 1 + 10569628158092 T + p^{23} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 225210108898504 T + p^{23} T^{2} )( 1 + 170735709367906 T + p^{23} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 102740902654734670 T + p^{23} T^{2} )( 1 + 102740902654734670 T + p^{23} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1714109869354088004 T + p^{23} T^{2} )( 1 + 1321635356841084986 T + p^{23} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 1323547242651585272 T + p^{23} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - \)\(13\!\cdots\!18\)\( T + p^{23} T^{2} )( 1 + 15848965445189145772 T + p^{23} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + \)\(65\!\cdots\!68\)\( T + p^{23} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{23} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - \)\(49\!\cdots\!98\)\( T + p^{23} T^{2} )( 1 - \)\(20\!\cdots\!08\)\( T + p^{23} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{23} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{46} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - \)\(14\!\cdots\!70\)\( T + p^{23} T^{2} )( 1 + \)\(14\!\cdots\!70\)\( T + p^{23} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - \)\(27\!\cdots\!84\)\( T + p^{23} T^{2} )( 1 + \)\(13\!\cdots\!06\)\( T + p^{23} T^{2} ) \) |
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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
−13.76202988421072047528300820307, −12.49496581159904484547465515224, −12.30094763620220182530898534474, −11.40067211342984686792088020012, −10.63911573537441566148389461657, −9.999286439589133238080878849055, −9.621112192974748213489242027870, −9.243782023474765081633944460558, −8.319090248414931155549109234847, −7.49372483702507535951822453281, −7.37446905588955853692425830775, −6.44421195977176481038429667956, −5.55226540583131124035545810780, −4.86477764536022706354853956683, −4.19852483922844013520535194392, −2.87725499505828398311044719697, −2.36076402591271731190370620899, −1.80732441899870000783268298188, −0.998725312743691065513465759030, −0.19129748815130489858023064839,
0.19129748815130489858023064839, 0.998725312743691065513465759030, 1.80732441899870000783268298188, 2.36076402591271731190370620899, 2.87725499505828398311044719697, 4.19852483922844013520535194392, 4.86477764536022706354853956683, 5.55226540583131124035545810780, 6.44421195977176481038429667956, 7.37446905588955853692425830775, 7.49372483702507535951822453281, 8.319090248414931155549109234847, 9.243782023474765081633944460558, 9.621112192974748213489242027870, 9.999286439589133238080878849055, 10.63911573537441566148389461657, 11.40067211342984686792088020012, 12.30094763620220182530898534474, 12.49496581159904484547465515224, 13.76202988421072047528300820307
