| L(s) = 1 | − 3-s + 1.41·5-s + 2·7-s + 9-s + 11-s + 13-s − 1.41·15-s − 3.41·17-s + 4.82·19-s − 2·21-s + 0.828·23-s − 2.99·25-s − 27-s − 9.07·29-s − 6.24·31-s − 33-s + 2.82·35-s − 11.6·37-s − 39-s − 1.17·41-s + 0.242·43-s + 1.41·45-s − 4.48·47-s − 3·49-s + 3.41·51-s + 2·53-s + 1.41·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.632·5-s + 0.755·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.365·15-s − 0.828·17-s + 1.10·19-s − 0.436·21-s + 0.172·23-s − 0.599·25-s − 0.192·27-s − 1.68·29-s − 1.12·31-s − 0.174·33-s + 0.478·35-s − 1.91·37-s − 0.160·39-s − 0.182·41-s + 0.0370·43-s + 0.210·45-s − 0.654·47-s − 0.428·49-s + 0.478·51-s + 0.274·53-s + 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 9.07T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 0.242T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.485T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 - 0.242T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−7.48893076607711952478434636659, −6.93642667619504502594997579755, −6.09498411682453723189603741944, −5.46004483086053394164649244294, −4.97578480432390573630353237433, −4.03950769278670799724098243709, −3.25630609288498386305117751841, −1.91075479554379781913261339442, −1.51849688131211854983026299909, 0,
1.51849688131211854983026299909, 1.91075479554379781913261339442, 3.25630609288498386305117751841, 4.03950769278670799724098243709, 4.97578480432390573630353237433, 5.46004483086053394164649244294, 6.09498411682453723189603741944, 6.93642667619504502594997579755, 7.48893076607711952478434636659
