| L(s) = 1 | − 2-s + 4-s + 2.87·7-s − 8-s − 2.40·11-s + 0.597·13-s − 2.87·14-s + 16-s − 2.87·17-s − 19-s + 2.40·22-s − 7.03·23-s − 0.597·26-s + 2.87·28-s + 6.43·29-s − 2.75·31-s − 32-s + 2.87·34-s + 2.80·37-s + 38-s − 6.27·41-s + 11.7·43-s − 2.40·44-s + 7.03·46-s + 3.08·47-s + 1.27·49-s + 0.597·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.08·7-s − 0.353·8-s − 0.724·11-s + 0.165·13-s − 0.769·14-s + 0.250·16-s − 0.697·17-s − 0.229·19-s + 0.512·22-s − 1.46·23-s − 0.117·26-s + 0.543·28-s + 1.19·29-s − 0.494·31-s − 0.176·32-s + 0.493·34-s + 0.460·37-s + 0.162·38-s − 0.980·41-s + 1.78·43-s − 0.362·44-s + 1.03·46-s + 0.449·47-s + 0.182·49-s + 0.0829·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + 2.40T + 11T^{2} \) |
| 13 | \( 1 - 0.597T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 23 | \( 1 + 7.03T + 23T^{2} \) |
| 29 | \( 1 - 6.43T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 - 2.80T + 37T^{2} \) |
| 41 | \( 1 + 6.27T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.08T + 47T^{2} \) |
| 53 | \( 1 - 4.96T + 53T^{2} \) |
| 59 | \( 1 - 1.15T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 + 3.83T + 73T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
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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
−7.60877745024817228850125989117, −6.93349269272557543653813498476, −6.07516545074461176908785445210, −5.47462439754472857292066248047, −4.58524986633065265987230769362, −3.98946906608135887535042921480, −2.74955159909069694727664918496, −2.12816060833544475627575071576, −1.23923983966876535160471253694, 0,
1.23923983966876535160471253694, 2.12816060833544475627575071576, 2.74955159909069694727664918496, 3.98946906608135887535042921480, 4.58524986633065265987230769362, 5.47462439754472857292066248047, 6.07516545074461176908785445210, 6.93349269272557543653813498476, 7.60877745024817228850125989117
