| L(s) = 1 | + 2·3-s + 7-s + 9-s + 2·13-s − 8·19-s + 2·21-s + 6·25-s − 4·27-s + 4·39-s + 6·43-s − 6·49-s − 16·57-s + 63-s + 24·67-s + 3·73-s + 12·75-s − 4·79-s − 11·81-s + 2·91-s + 16·97-s + 20·103-s − 20·109-s + 2·117-s − 14·121-s + 127-s + 12·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.83·19-s + 0.436·21-s + 6/5·25-s − 0.769·27-s + 0.640·39-s + 0.914·43-s − 6/7·49-s − 2.11·57-s + 0.125·63-s + 2.93·67-s + 0.351·73-s + 1.38·75-s − 0.450·79-s − 1.22·81-s + 0.209·91-s + 1.62·97-s + 1.97·103-s − 1.91·109-s + 0.184·117-s − 1.27·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1177344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1177344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.146761764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.146761764\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.239336093449914678215428399846, −7.77088674661830994616706802953, −7.22073181493104017937857231333, −6.73874007584629692411494077417, −6.39304485958783668461557447120, −5.88928065681144139011770811900, −5.32892045406048220080698236465, −4.79080982183336146712261414882, −4.29402601048030716451673073527, −3.85696710448615271195762637707, −3.33141183645955123887138028271, −2.78794560387640188810932761881, −2.18144682302881447475320892165, −1.78754946425480483953537519008, −0.73688601445894288440962212577,
0.73688601445894288440962212577, 1.78754946425480483953537519008, 2.18144682302881447475320892165, 2.78794560387640188810932761881, 3.33141183645955123887138028271, 3.85696710448615271195762637707, 4.29402601048030716451673073527, 4.79080982183336146712261414882, 5.32892045406048220080698236465, 5.88928065681144139011770811900, 6.39304485958783668461557447120, 6.73874007584629692411494077417, 7.22073181493104017937857231333, 7.77088674661830994616706802953, 8.239336093449914678215428399846
