| L(s) = 1 | − 3-s + 9-s − 3·11-s + 5·19-s + 25-s − 27-s + 3·33-s + 15·41-s − 7·43-s + 5·49-s − 5·57-s + 3·59-s + 8·67-s + 4·73-s − 75-s + 81-s + 15·83-s + 18·89-s − 11·97-s − 3·99-s − 9·107-s + 24·113-s + 5·121-s − 15·123-s + 127-s + 7·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.904·11-s + 1.14·19-s + 1/5·25-s − 0.192·27-s + 0.522·33-s + 2.34·41-s − 1.06·43-s + 5/7·49-s − 0.662·57-s + 0.390·59-s + 0.977·67-s + 0.468·73-s − 0.115·75-s + 1/9·81-s + 1.64·83-s + 1.90·89-s − 1.11·97-s − 0.301·99-s − 0.870·107-s + 2.25·113-s + 5/11·121-s − 1.35·123-s + 0.0887·127-s + 0.616·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.622511463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.622511463\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.79120924551007060693427110935, −7.28918786276887823799767762989, −6.93076623251852629842100645373, −6.47124481595403738606107107164, −5.94356561039630516088964153644, −5.55182253160799831985479889101, −5.27682873501599039200724832301, −4.67038464362361036612458762746, −4.42990045130904662476792329892, −3.56017108850573248747822124778, −3.38256733223392825213409861337, −2.50326146026616319145866569706, −2.21799236789061802657410245959, −1.21751099379307371715091888576, −0.59381258230769200426858685129,
0.59381258230769200426858685129, 1.21751099379307371715091888576, 2.21799236789061802657410245959, 2.50326146026616319145866569706, 3.38256733223392825213409861337, 3.56017108850573248747822124778, 4.42990045130904662476792329892, 4.67038464362361036612458762746, 5.27682873501599039200724832301, 5.55182253160799831985479889101, 5.94356561039630516088964153644, 6.47124481595403738606107107164, 6.93076623251852629842100645373, 7.28918786276887823799767762989, 7.79120924551007060693427110935
