| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 1.73·11-s − 1.46·13-s − 14-s + 16-s + 4·17-s + 2.26·19-s − 1.73·22-s − 4.46·23-s − 1.46·26-s − 28-s − 5.46·29-s − 3.73·31-s + 32-s + 4·34-s + 7.19·37-s + 2.26·38-s + 9.92·41-s − 11.4·43-s − 1.73·44-s − 4.46·46-s + 4.53·47-s + 49-s − 1.46·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.377·7-s + 0.353·8-s − 0.522·11-s − 0.406·13-s − 0.267·14-s + 0.250·16-s + 0.970·17-s + 0.520·19-s − 0.369·22-s − 0.930·23-s − 0.287·26-s − 0.188·28-s − 1.01·29-s − 0.670·31-s + 0.176·32-s + 0.685·34-s + 1.18·37-s + 0.367·38-s + 1.55·41-s − 1.74·43-s − 0.261·44-s − 0.658·46-s + 0.661·47-s + 0.142·49-s − 0.203·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.917758121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917758121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 + 4.46T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−7.59457303358246197639567967723, −7.04926916880187580824597265801, −6.15194543163984340020471297311, −5.57700500997583406162032331441, −5.08730477168330522846237023405, −4.07867617627767071646391605737, −3.55901093834034287860879105950, −2.69188879442488222906320101018, −1.98002411848903735591455501251, −0.71731011929297145050631491686,
0.71731011929297145050631491686, 1.98002411848903735591455501251, 2.69188879442488222906320101018, 3.55901093834034287860879105950, 4.07867617627767071646391605737, 5.08730477168330522846237023405, 5.57700500997583406162032331441, 6.15194543163984340020471297311, 7.04926916880187580824597265801, 7.59457303358246197639567967723
