| L(s) = 1 | + 2·3-s + 3·9-s + 8·11-s + 4·17-s + 8·19-s − 6·25-s + 4·27-s + 16·33-s − 12·41-s + 8·43-s + 49-s + 8·51-s + 16·57-s − 8·59-s − 8·67-s + 20·73-s − 12·75-s + 5·81-s + 8·83-s − 12·89-s − 28·97-s + 24·99-s − 24·107-s − 28·113-s + 26·121-s − 24·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 2.41·11-s + 0.970·17-s + 1.83·19-s − 6/5·25-s + 0.769·27-s + 2.78·33-s − 1.87·41-s + 1.21·43-s + 1/7·49-s + 1.12·51-s + 2.11·57-s − 1.04·59-s − 0.977·67-s + 2.34·73-s − 1.38·75-s + 5/9·81-s + 0.878·83-s − 1.27·89-s − 2.84·97-s + 2.41·99-s − 2.32·107-s − 2.63·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.875464647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.875464647\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.579197535720292899748335278003, −7.963562405403294064353399542187, −7.941030481552254497163330431349, −7.14585602865319249961807679325, −6.89499765960031010222506198709, −6.43557747993001049134741297977, −5.69761771888975023011122238972, −5.40100953109829334895217907278, −4.54031360247962128301261767712, −4.04752481141371955267217260485, −3.54052974946578322805924483948, −3.29683272363669349411451437037, −2.47888758091438588042078740356, −1.50800619952171259085588630937, −1.23111713389498587746226154587,
1.23111713389498587746226154587, 1.50800619952171259085588630937, 2.47888758091438588042078740356, 3.29683272363669349411451437037, 3.54052974946578322805924483948, 4.04752481141371955267217260485, 4.54031360247962128301261767712, 5.40100953109829334895217907278, 5.69761771888975023011122238972, 6.43557747993001049134741297977, 6.89499765960031010222506198709, 7.14585602865319249961807679325, 7.941030481552254497163330431349, 7.963562405403294064353399542187, 8.579197535720292899748335278003
