| L(s) = 1 | − 2·3-s + 4-s + 5-s + 8·7-s + 3·9-s − 2·12-s − 2·15-s + 16-s − 6·17-s − 8·19-s + 20-s − 16·21-s + 25-s − 4·27-s + 8·28-s + 8·35-s + 3·36-s − 4·37-s + 3·45-s − 2·48-s + 34·49-s + 12·51-s + 16·57-s − 2·60-s + 24·63-s + 64-s − 6·68-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.447·5-s + 3.02·7-s + 9-s − 0.577·12-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 1.83·19-s + 0.223·20-s − 3.49·21-s + 1/5·25-s − 0.769·27-s + 1.51·28-s + 1.35·35-s + 1/2·36-s − 0.657·37-s + 0.447·45-s − 0.288·48-s + 34/7·49-s + 1.68·51-s + 2.11·57-s − 0.258·60-s + 3.02·63-s + 1/8·64-s − 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.437334552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437334552\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.79711986387068141575976779625, −7.56689807773085737112376729195, −7.19170400608151927655187626887, −6.38949936725272530282824625186, −6.37150043364000269525301859690, −5.82487279118675041342104526304, −5.07079211702149946545201877286, −5.01944387438421311944798675589, −4.46472359494342999265261204391, −4.30912662284802770612117339996, −3.45323495695880355625020856878, −2.19891095670466256908754466292, −2.07452981833800877942993394416, −1.67422833535095353270525529506, −0.74314124481069084021545695805,
0.74314124481069084021545695805, 1.67422833535095353270525529506, 2.07452981833800877942993394416, 2.19891095670466256908754466292, 3.45323495695880355625020856878, 4.30912662284802770612117339996, 4.46472359494342999265261204391, 5.01944387438421311944798675589, 5.07079211702149946545201877286, 5.82487279118675041342104526304, 6.37150043364000269525301859690, 6.38949936725272530282824625186, 7.19170400608151927655187626887, 7.56689807773085737112376729195, 7.79711986387068141575976779625
