| L(s) = 1 | − 5-s − 9-s + 2·11-s + 2·31-s + 45-s − 2·49-s − 2·55-s + 2·59-s − 2·71-s + 2·89-s − 2·99-s + 3·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 5-s − 9-s + 2·11-s + 2·31-s + 45-s − 2·49-s − 2·55-s + 2·59-s − 2·71-s + 2·89-s − 2·99-s + 3·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.120697584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120697584\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.075751851269258745975030761993, −8.518319594994623319076389768314, −8.331043546101537887068527858373, −7.79551053844817427088859375420, −7.59749879773738675976605309839, −7.02429305130262922309361396371, −6.66598325762338124525744882663, −6.40027671107294541287041652865, −6.02680022567003841989920176188, −5.68429494620148868538969595951, −5.03543511034848864512015136990, −4.64775630912575104777832088094, −4.35135847694213903017854313937, −3.76950079489309455784339537086, −3.62597990196896818456932551952, −3.03809788939863623949091890353, −2.67715785801943253071926931538, −1.94097323561496229679135522449, −1.35681655633745390270161755919, −0.67267543587159371333508382746,
0.67267543587159371333508382746, 1.35681655633745390270161755919, 1.94097323561496229679135522449, 2.67715785801943253071926931538, 3.03809788939863623949091890353, 3.62597990196896818456932551952, 3.76950079489309455784339537086, 4.35135847694213903017854313937, 4.64775630912575104777832088094, 5.03543511034848864512015136990, 5.68429494620148868538969595951, 6.02680022567003841989920176188, 6.40027671107294541287041652865, 6.66598325762338124525744882663, 7.02429305130262922309361396371, 7.59749879773738675976605309839, 7.79551053844817427088859375420, 8.331043546101537887068527858373, 8.518319594994623319076389768314, 9.075751851269258745975030761993
