| L(s) = 1 | + 0.772·2-s − 1.40·4-s − 5-s + 1.62·7-s − 2.62·8-s − 0.772·10-s − 4.80·13-s + 1.25·14-s + 0.772·16-s + 2.17·17-s + 5.17·19-s + 1.40·20-s − 4.62·23-s + 25-s − 3.71·26-s − 2.28·28-s + 2.45·29-s + 5.80·31-s + 5.85·32-s + 1.68·34-s − 1.62·35-s + 8.88·37-s + 4·38-s + 2.62·40-s − 5.09·41-s − 1.54·43-s − 3.57·46-s + ⋯ |
| L(s) = 1 | + 0.546·2-s − 0.701·4-s − 0.447·5-s + 0.616·7-s − 0.929·8-s − 0.244·10-s − 1.33·13-s + 0.336·14-s + 0.193·16-s + 0.527·17-s + 1.18·19-s + 0.313·20-s − 0.965·23-s + 0.200·25-s − 0.728·26-s − 0.432·28-s + 0.455·29-s + 1.04·31-s + 1.03·32-s + 0.288·34-s − 0.275·35-s + 1.46·37-s + 0.648·38-s + 0.415·40-s − 0.795·41-s − 0.235·43-s − 0.527·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.772T + 2T^{2} \) |
| 7 | \( 1 - 1.62T + 7T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 - 8.88T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 + 1.54T + 43T^{2} \) |
| 47 | \( 1 + 9.72T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 4.72T + 97T^{2} \) |
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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
−7.979682407620983258205508023123, −7.18311412156709596113941036948, −6.22051409707319989732536995794, −5.41813135528031197441433374113, −4.78267052133339397383141228919, −4.32716532626397387796716312982, −3.33068655945813211326110106340, −2.64584457251962317860744735760, −1.26407724166675357876117583778, 0,
1.26407724166675357876117583778, 2.64584457251962317860744735760, 3.33068655945813211326110106340, 4.32716532626397387796716312982, 4.78267052133339397383141228919, 5.41813135528031197441433374113, 6.22051409707319989732536995794, 7.18311412156709596113941036948, 7.979682407620983258205508023123
