| L(s) = 1 | + 2-s + 3-s − 4-s + 4·5-s + 6-s − 3·8-s + 9-s + 4·10-s − 12-s + 4·15-s − 16-s + 18-s + 8·19-s − 4·20-s − 3·24-s + 2·25-s + 27-s + 4·29-s + 4·30-s + 5·32-s − 36-s + 8·38-s − 12·40-s − 8·43-s + 4·45-s − 48-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 1.78·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s + 1.03·15-s − 1/4·16-s + 0.235·18-s + 1.83·19-s − 0.894·20-s − 0.612·24-s + 2/5·25-s + 0.192·27-s + 0.742·29-s + 0.730·30-s + 0.883·32-s − 1/6·36-s + 1.29·38-s − 1.89·40-s − 1.21·43-s + 0.596·45-s − 0.144·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.981749119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.981749119\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.465508899880048812015346580650, −9.457705101580370517739336150901, −8.900585072204336118527974818427, −8.202224728141808387663165849679, −7.81295430367747747098376616892, −7.02967607454599654192884435004, −6.43796562380819911728572291466, −5.94607665445287604157300008429, −5.46214860425768039190452613646, −5.02709416296405066539370067618, −4.41591285682058366018514470855, −3.50387103718831454239019194187, −3.05422074105458389777226041971, −2.25227517080335123972207039631, −1.36358574500507065097493451168,
1.36358574500507065097493451168, 2.25227517080335123972207039631, 3.05422074105458389777226041971, 3.50387103718831454239019194187, 4.41591285682058366018514470855, 5.02709416296405066539370067618, 5.46214860425768039190452613646, 5.94607665445287604157300008429, 6.43796562380819911728572291466, 7.02967607454599654192884435004, 7.81295430367747747098376616892, 8.202224728141808387663165849679, 8.900585072204336118527974818427, 9.457705101580370517739336150901, 9.465508899880048812015346580650
