| L(s) = 1 | − 3·4-s − 3·9-s + 5·16-s + 25-s + 16·31-s + 9·36-s + 4·37-s + 14·49-s − 3·64-s + 32·67-s + 9·81-s + 20·97-s − 3·100-s − 8·103-s − 48·124-s + 127-s + 131-s + 137-s + 139-s − 15·144-s − 12·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 9-s + 5/4·16-s + 1/5·25-s + 2.87·31-s + 3/2·36-s + 0.657·37-s + 2·49-s − 3/8·64-s + 3.90·67-s + 81-s + 2.03·97-s − 0.299·100-s − 0.788·103-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/4·144-s − 0.986·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.486936948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486936948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.78728288752897792279316403260, −7.01093662002827258333914006866, −6.67080600433344938839680502667, −6.20754250123106621176826580962, −5.75093741871164811723096969890, −5.38927227596219584800975814596, −4.86240608307875663974074455865, −4.67393357900988723058463796360, −4.00437885023502522077289730463, −3.80099383791037569282844736145, −3.06719206473458846961661923692, −2.64280845356380287461338230828, −2.10148632655396229867010033896, −0.882695747465533974580705371392, −0.65044647522797037357101848717,
0.65044647522797037357101848717, 0.882695747465533974580705371392, 2.10148632655396229867010033896, 2.64280845356380287461338230828, 3.06719206473458846961661923692, 3.80099383791037569282844736145, 4.00437885023502522077289730463, 4.67393357900988723058463796360, 4.86240608307875663974074455865, 5.38927227596219584800975814596, 5.75093741871164811723096969890, 6.20754250123106621176826580962, 6.67080600433344938839680502667, 7.01093662002827258333914006866, 7.78728288752897792279316403260
