| L(s) = 1 | − 1.91·2-s − 3.05·3-s + 1.66·4-s + 5.84·6-s − 2.87·7-s + 0.633·8-s + 6.32·9-s − 1.47·11-s − 5.09·12-s − 3.05·13-s + 5.51·14-s − 4.55·16-s + 5.66·17-s − 12.1·18-s − 8.00·19-s + 8.79·21-s + 2.82·22-s − 2.12·23-s − 1.93·24-s + 5.84·26-s − 10.1·27-s − 4.80·28-s + 3.25·29-s + 8.90·31-s + 7.45·32-s + 4.49·33-s − 10.8·34-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 1.76·3-s + 0.834·4-s + 2.38·6-s − 1.08·7-s + 0.224·8-s + 2.10·9-s − 0.444·11-s − 1.47·12-s − 0.846·13-s + 1.47·14-s − 1.13·16-s + 1.37·17-s − 2.85·18-s − 1.83·19-s + 1.91·21-s + 0.601·22-s − 0.444·23-s − 0.394·24-s + 1.14·26-s − 1.95·27-s − 0.908·28-s + 0.605·29-s + 1.59·31-s + 1.31·32-s + 0.783·33-s − 1.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 + 3.05T + 3T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 + 8.00T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 3.16T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 + 9.29T + 59T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 4.36T + 89T^{2} \) |
| 97 | \( 1 - 2.76T + 97T^{2} \) |
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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
−9.465417478865391235457362913461, −8.167219074481962328667547085839, −7.52029272368176214030495580597, −6.49336037688736782692727946478, −6.19180777908115661585512760106, −5.02620966356084689058906747573, −4.20632671785508086963330364192, −2.49519689997978099350329364308, −0.940179281939347153737498340676, 0,
0.940179281939347153737498340676, 2.49519689997978099350329364308, 4.20632671785508086963330364192, 5.02620966356084689058906747573, 6.19180777908115661585512760106, 6.49336037688736782692727946478, 7.52029272368176214030495580597, 8.167219074481962328667547085839, 9.465417478865391235457362913461
