| L(s) = 1 | + 3-s − 0.882·5-s − 2.85·7-s + 9-s + 2.43·11-s + 1.92·13-s − 0.882·15-s − 2.82·17-s + 5.97·19-s − 2.85·21-s − 3.88·23-s − 4.22·25-s + 27-s − 6.73·29-s + 0.533·31-s + 2.43·33-s + 2.52·35-s + 3.20·37-s + 1.92·39-s − 2.13·41-s − 7.27·43-s − 0.882·45-s − 0.965·47-s + 1.16·49-s − 2.82·51-s + 11.3·53-s − 2.15·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.394·5-s − 1.07·7-s + 0.333·9-s + 0.734·11-s + 0.533·13-s − 0.227·15-s − 0.686·17-s + 1.37·19-s − 0.623·21-s − 0.809·23-s − 0.844·25-s + 0.192·27-s − 1.25·29-s + 0.0958·31-s + 0.424·33-s + 0.426·35-s + 0.526·37-s + 0.308·39-s − 0.334·41-s − 1.11·43-s − 0.131·45-s − 0.140·47-s + 0.165·49-s − 0.396·51-s + 1.55·53-s − 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 0.882T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 1.92T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 - 0.533T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 2.13T + 41T^{2} \) |
| 43 | \( 1 + 7.27T + 43T^{2} \) |
| 47 | \( 1 + 0.965T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 8.38T + 59T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 3.91T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 9.34T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67283003193799514976341309443, −7.11070067145819192630483983602, −6.32928611636437863308298245710, −5.76187164257758031104052865215, −4.66106230809695522496786345623, −3.64552032806354457901921621983, −3.55002711205485855977922294722, −2.40955826455452552664092931560, −1.37410577090458877680436467261, 0,
1.37410577090458877680436467261, 2.40955826455452552664092931560, 3.55002711205485855977922294722, 3.64552032806354457901921621983, 4.66106230809695522496786345623, 5.76187164257758031104052865215, 6.32928611636437863308298245710, 7.11070067145819192630483983602, 7.67283003193799514976341309443
