| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·9-s + 5·16-s + 4·18-s + 16·23-s − 6·25-s − 6·32-s − 6·36-s − 4·43-s − 32·46-s − 7·49-s + 12·50-s + 12·53-s + 7·64-s − 16·67-s + 16·71-s + 8·72-s + 16·79-s − 5·81-s + 8·86-s + 48·92-s + 14·98-s − 18·100-s − 24·106-s + 16·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s + 5/4·16-s + 0.942·18-s + 3.33·23-s − 6/5·25-s − 1.06·32-s − 36-s − 0.609·43-s − 4.71·46-s − 49-s + 1.69·50-s + 1.64·53-s + 7/8·64-s − 1.95·67-s + 1.89·71-s + 0.942·72-s + 1.80·79-s − 5/9·81-s + 0.862·86-s + 5.00·92-s + 1.41·98-s − 9/5·100-s − 2.33·106-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1552516 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1552516 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9103095416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9103095416\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.77948089534178376910067015471, −7.62278173895024006072090672250, −7.17734181742489309105923485780, −6.67854632460910561231510247828, −6.39157448299470323306279432463, −5.86058565685324849345856116979, −5.18755879988201558728632555274, −5.10345918500099198340016282816, −4.29555192736502971554821928846, −3.49313221834977534889859645155, −3.16476680941726517752422559781, −2.60554978310542183468446192843, −2.01056152480496856645260763012, −1.24613795354549553370877097103, −0.55881487121039629178406292495,
0.55881487121039629178406292495, 1.24613795354549553370877097103, 2.01056152480496856645260763012, 2.60554978310542183468446192843, 3.16476680941726517752422559781, 3.49313221834977534889859645155, 4.29555192736502971554821928846, 5.10345918500099198340016282816, 5.18755879988201558728632555274, 5.86058565685324849345856116979, 6.39157448299470323306279432463, 6.67854632460910561231510247828, 7.17734181742489309105923485780, 7.62278173895024006072090672250, 7.77948089534178376910067015471
