| L(s) = 1 | − 3-s + 9-s + 12·13-s + 8·19-s + 25-s − 27-s + 16·31-s − 4·37-s − 12·39-s − 24·43-s − 14·49-s − 8·57-s + 28·61-s − 8·67-s − 12·73-s − 75-s + 16·79-s + 81-s − 16·93-s + 4·97-s − 36·109-s + 4·111-s + 12·117-s − 6·121-s + 127-s + 24·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 3.32·13-s + 1.83·19-s + 1/5·25-s − 0.192·27-s + 2.87·31-s − 0.657·37-s − 1.92·39-s − 3.65·43-s − 2·49-s − 1.05·57-s + 3.58·61-s − 0.977·67-s − 1.40·73-s − 0.115·75-s + 1.80·79-s + 1/9·81-s − 1.65·93-s + 0.406·97-s − 3.44·109-s + 0.379·111-s + 1.10·117-s − 0.545·121-s + 0.0887·127-s + 2.11·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.768510500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.768510500\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−9.234498555197333169762430563636, −8.372242663067272040741337380641, −8.364107116024692137216049760263, −8.062184279727733628805054904271, −6.97039739431553724911307537358, −6.71560882928791827542770443897, −6.23234003008114203357336027651, −5.86178029126505651814373374846, −5.06868408963538865423952877412, −4.86312932086210572949074191291, −3.76264729663280098898303229481, −3.57148112685573562185271725821, −2.89778258507562116298634418143, −1.48909737355432343830877670544, −1.08829539378793470967143951854,
1.08829539378793470967143951854, 1.48909737355432343830877670544, 2.89778258507562116298634418143, 3.57148112685573562185271725821, 3.76264729663280098898303229481, 4.86312932086210572949074191291, 5.06868408963538865423952877412, 5.86178029126505651814373374846, 6.23234003008114203357336027651, 6.71560882928791827542770443897, 6.97039739431553724911307537358, 8.062184279727733628805054904271, 8.364107116024692137216049760263, 8.372242663067272040741337380641, 9.234498555197333169762430563636
