| L(s) = 1 | − 5-s + 8·11-s + 13-s + 3·17-s − 8·19-s + 5·23-s + 5·25-s + 7·29-s + 8·31-s + 20·37-s − 5·41-s + 5·43-s − 7·47-s − 14·49-s + 11·53-s − 8·55-s + 3·59-s − 11·61-s − 65-s + 3·67-s + 11·71-s − 15·73-s + 13·79-s − 3·85-s + 3·89-s + 8·95-s + 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 2.41·11-s + 0.277·13-s + 0.727·17-s − 1.83·19-s + 1.04·23-s + 25-s + 1.29·29-s + 1.43·31-s + 3.28·37-s − 0.780·41-s + 0.762·43-s − 1.02·47-s − 2·49-s + 1.51·53-s − 1.07·55-s + 0.390·59-s − 1.40·61-s − 0.124·65-s + 0.366·67-s + 1.30·71-s − 1.75·73-s + 1.46·79-s − 0.325·85-s + 0.317·89-s + 0.820·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330183810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330183810\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 11 T + 50 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65176902643979957407562522284, −10.39055879973203081217386114195, −9.611260232005024940752460231106, −9.497351295310734943671927488449, −8.952138043441643345854234310768, −8.580652816347476968760742032571, −8.015615020936770040446543523647, −7.933657381882974717275263284623, −6.94265622070160731094303134763, −6.65806425345277821124763210627, −6.30556778419459815176130522348, −6.12993635719395679725819795471, −5.05780718076940847184524796367, −4.68464939759918247971089997218, −4.02060887575246387022468094790, −3.96870431994406708542770645362, −2.98585148995980340121253393424, −2.59662526913971519372896036577, −1.37326365594736512890603353636, −0.977606918264635694411434422851,
0.977606918264635694411434422851, 1.37326365594736512890603353636, 2.59662526913971519372896036577, 2.98585148995980340121253393424, 3.96870431994406708542770645362, 4.02060887575246387022468094790, 4.68464939759918247971089997218, 5.05780718076940847184524796367, 6.12993635719395679725819795471, 6.30556778419459815176130522348, 6.65806425345277821124763210627, 6.94265622070160731094303134763, 7.933657381882974717275263284623, 8.015615020936770040446543523647, 8.580652816347476968760742032571, 8.952138043441643345854234310768, 9.497351295310734943671927488449, 9.611260232005024940752460231106, 10.39055879973203081217386114195, 10.65176902643979957407562522284
