| L(s) = 1 | + 3·3-s − 4·7-s + 6·9-s − 6·19-s − 12·21-s + 9·27-s + 3·31-s + 37-s − 26·43-s + 9·49-s − 18·57-s + 15·61-s − 24·63-s − 16·67-s + 27·73-s + 13·79-s + 9·81-s + 9·93-s − 33·103-s + 2·109-s + 3·111-s − 11·121-s + 127-s − 78·129-s + 131-s + 24·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.51·7-s + 2·9-s − 1.37·19-s − 2.61·21-s + 1.73·27-s + 0.538·31-s + 0.164·37-s − 3.96·43-s + 9/7·49-s − 2.38·57-s + 1.92·61-s − 3.02·63-s − 1.95·67-s + 3.16·73-s + 1.46·79-s + 81-s + 0.933·93-s − 3.25·103-s + 0.191·109-s + 0.284·111-s − 121-s + 0.0887·127-s − 6.86·129-s + 0.0873·131-s + 2.08·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.721224573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721224573\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.427166226359367537511948221315, −8.661724395501958977729491669741, −8.658800311390515134522224373433, −8.224772261083050511153160437764, −7.954247371253177825668735956615, −7.40052455641688474214672250805, −6.86072647951001461470110821494, −6.58243371987730664460637482408, −6.52068835648759771299220985608, −5.85448920140520726094720335725, −5.16528651561442689408882903949, −4.84418199301923632997843338166, −4.17659702734318322918859277210, −3.77652796553541269576553591449, −3.50203604629021938684752076343, −2.98811235638487877318818563393, −2.63318211974073032875383706191, −2.03649988431347988753783819853, −1.60237377675957337527355375719, −0.49510254617224935963253262031,
0.49510254617224935963253262031, 1.60237377675957337527355375719, 2.03649988431347988753783819853, 2.63318211974073032875383706191, 2.98811235638487877318818563393, 3.50203604629021938684752076343, 3.77652796553541269576553591449, 4.17659702734318322918859277210, 4.84418199301923632997843338166, 5.16528651561442689408882903949, 5.85448920140520726094720335725, 6.52068835648759771299220985608, 6.58243371987730664460637482408, 6.86072647951001461470110821494, 7.40052455641688474214672250805, 7.954247371253177825668735956615, 8.224772261083050511153160437764, 8.658800311390515134522224373433, 8.661724395501958977729491669741, 9.427166226359367537511948221315
