| L(s) = 1 | − 3-s + 3.23·5-s − 7-s + 9-s − 2·11-s − 3.23·13-s − 3.23·15-s − 6.47·17-s + 19-s + 21-s + 7.70·23-s + 5.47·25-s − 27-s − 4.47·29-s + 0.763·31-s + 2·33-s − 3.23·35-s + 5.23·37-s + 3.23·39-s + 2·41-s + 10.4·43-s + 3.23·45-s − 9.70·47-s + 49-s + 6.47·51-s + 8.47·53-s − 6.47·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.44·5-s − 0.377·7-s + 0.333·9-s − 0.603·11-s − 0.897·13-s − 0.835·15-s − 1.56·17-s + 0.229·19-s + 0.218·21-s + 1.60·23-s + 1.09·25-s − 0.192·27-s − 0.830·29-s + 0.137·31-s + 0.348·33-s − 0.546·35-s + 0.860·37-s + 0.518·39-s + 0.312·41-s + 1.59·43-s + 0.482·45-s − 1.41·47-s + 0.142·49-s + 0.906·51-s + 1.16·53-s − 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 0.763T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.36669059121803215735298991578, −6.94794024092435305733205871937, −6.14329579347987773826354802434, −5.61956195964614208478148994423, −4.93622076675538860290802181389, −4.28099331642809708863295151047, −2.85623194208018120783506231031, −2.38566800926723974513394845997, −1.34357066211476855673484280248, 0,
1.34357066211476855673484280248, 2.38566800926723974513394845997, 2.85623194208018120783506231031, 4.28099331642809708863295151047, 4.93622076675538860290802181389, 5.61956195964614208478148994423, 6.14329579347987773826354802434, 6.94794024092435305733205871937, 7.36669059121803215735298991578
