| L(s) = 1 | − 4·5-s + 6·7-s − 8·11-s − 5·13-s + 8·19-s + 4·23-s + 5·25-s − 8·29-s − 2·31-s − 24·35-s − 10·37-s − 8·41-s − 5·43-s − 8·47-s + 13·49-s − 4·53-s + 32·55-s + 12·59-s + 61-s + 20·65-s + 3·67-s + 16·71-s + 15·73-s − 48·77-s − 7·79-s + 12·89-s − 30·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2.26·7-s − 2.41·11-s − 1.38·13-s + 1.83·19-s + 0.834·23-s + 25-s − 1.48·29-s − 0.359·31-s − 4.05·35-s − 1.64·37-s − 1.24·41-s − 0.762·43-s − 1.16·47-s + 13/7·49-s − 0.549·53-s + 4.31·55-s + 1.56·59-s + 0.128·61-s + 2.48·65-s + 0.366·67-s + 1.89·71-s + 1.75·73-s − 5.47·77-s − 0.787·79-s + 1.27·89-s − 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4621865418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4621865418\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 16 T + 185 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.746604429847896594359142399062, −8.379771143879936210409614059627, −8.092798621716948607595969057627, −7.931069722793744539918073359351, −7.54583335170704073171108238188, −7.37187026073386030937008334899, −7.09551689925654180195612114733, −6.53259120101027644689625391772, −5.45098654943333966895459852095, −5.32788757600972652638742002583, −5.06493577267858855801850465563, −4.92027892878213549974287614787, −4.50116185918283351293819646895, −3.57755984016781084058066171680, −3.56781055482673680340701383548, −3.00173730888348953393402973878, −2.12309838856867539168310622014, −2.08450207806930729844466128052, −1.17250722444305766221799270435, −0.23297870543730096797677302709,
0.23297870543730096797677302709, 1.17250722444305766221799270435, 2.08450207806930729844466128052, 2.12309838856867539168310622014, 3.00173730888348953393402973878, 3.56781055482673680340701383548, 3.57755984016781084058066171680, 4.50116185918283351293819646895, 4.92027892878213549974287614787, 5.06493577267858855801850465563, 5.32788757600972652638742002583, 5.45098654943333966895459852095, 6.53259120101027644689625391772, 7.09551689925654180195612114733, 7.37187026073386030937008334899, 7.54583335170704073171108238188, 7.931069722793744539918073359351, 8.092798621716948607595969057627, 8.379771143879936210409614059627, 8.746604429847896594359142399062
