| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−7.71186497925010243460438877854, −7.07210472980103811211983505436, −6.01540342546925623228279012825, −5.32434737095783305198806645257, −4.68892818634659626201777614856, −3.69809676168292337458488503030, −3.07757290967532801326257403068, −2.16791010395709344372487601855, −1.69422163749019850652635472447, 0,
1.69422163749019850652635472447, 2.16791010395709344372487601855, 3.07757290967532801326257403068, 3.69809676168292337458488503030, 4.68892818634659626201777614856, 5.32434737095783305198806645257, 6.01540342546925623228279012825, 7.07210472980103811211983505436, 7.71186497925010243460438877854
