| L(s) = 1 | − 1.98·3-s + 0.688·5-s − 1.63·7-s + 0.950·9-s + 1.13·11-s − 4.69·13-s − 1.36·15-s + 3.66·17-s − 0.478·19-s + 3.25·21-s + 4.93·23-s − 4.52·25-s + 4.07·27-s + 3.04·29-s + 7.78·31-s − 2.25·33-s − 1.12·35-s − 1.05·37-s + 9.32·39-s − 10.5·41-s − 3.06·43-s + 0.654·45-s + 11.5·47-s − 4.32·49-s − 7.28·51-s + 6.59·53-s + 0.781·55-s + ⋯ |
| L(s) = 1 | − 1.14·3-s + 0.307·5-s − 0.618·7-s + 0.316·9-s + 0.342·11-s − 1.30·13-s − 0.353·15-s + 0.888·17-s − 0.109·19-s + 0.709·21-s + 1.02·23-s − 0.905·25-s + 0.783·27-s + 0.565·29-s + 1.39·31-s − 0.392·33-s − 0.190·35-s − 0.174·37-s + 1.49·39-s − 1.65·41-s − 0.468·43-s + 0.0976·45-s + 1.68·47-s − 0.617·49-s − 1.01·51-s + 0.905·53-s + 0.105·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 5 | \( 1 - 0.688T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 - 3.66T + 17T^{2} \) |
| 19 | \( 1 + 0.478T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 + 1.05T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 + 9.43T + 61T^{2} \) |
| 67 | \( 1 + 8.15T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 0.930T + 79T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 - 1.82T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.988812251150592739673246997341, −7.10013371753801514500138030362, −6.54416033206738434853888040204, −5.85677742261699154320373674108, −5.16183815246710402152289538912, −4.55357518434786837645987652474, −3.35708792914607424368747744130, −2.51560568798415704212588206867, −1.15435026218681620969323179994, 0,
1.15435026218681620969323179994, 2.51560568798415704212588206867, 3.35708792914607424368747744130, 4.55357518434786837645987652474, 5.16183815246710402152289538912, 5.85677742261699154320373674108, 6.54416033206738434853888040204, 7.10013371753801514500138030362, 7.988812251150592739673246997341
