| L(s) = 1 | − 2·3-s + 3·9-s + 2·11-s + 12·17-s + 16·19-s − 6·25-s − 4·27-s − 4·33-s − 20·41-s + 16·43-s + 2·49-s − 24·51-s − 32·57-s − 8·59-s + 24·67-s + 4·73-s + 12·75-s + 5·81-s + 24·83-s − 12·89-s + 4·97-s + 6·99-s − 24·107-s + 20·113-s + 3·121-s + 40·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.603·11-s + 2.91·17-s + 3.67·19-s − 6/5·25-s − 0.769·27-s − 0.696·33-s − 3.12·41-s + 2.43·43-s + 2/7·49-s − 3.36·51-s − 4.23·57-s − 1.04·59-s + 2.93·67-s + 0.468·73-s + 1.38·75-s + 5/9·81-s + 2.63·83-s − 1.27·89-s + 0.406·97-s + 0.603·99-s − 2.32·107-s + 1.88·113-s + 3/11·121-s + 3.60·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015927381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015927381\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.024470565850335561462456684411, −7.56089293819005546804727949953, −7.19819071506851382078022350491, −6.82430943807879406400182869542, −6.19015485249586140740060842128, −5.56993690843411272942663805412, −5.46405190258323470080919041481, −5.27293036458461851477097267556, −4.57486609696928672939637526621, −3.76305426337925729255035834153, −3.38108759817336471336235218518, −3.16578625772066727003923663964, −1.96689570329397719651168220011, −1.10197409729490824142182868653, −0.918992040659386671383695016650,
0.918992040659386671383695016650, 1.10197409729490824142182868653, 1.96689570329397719651168220011, 3.16578625772066727003923663964, 3.38108759817336471336235218518, 3.76305426337925729255035834153, 4.57486609696928672939637526621, 5.27293036458461851477097267556, 5.46405190258323470080919041481, 5.56993690843411272942663805412, 6.19015485249586140740060842128, 6.82430943807879406400182869542, 7.19819071506851382078022350491, 7.56089293819005546804727949953, 8.024470565850335561462456684411
