| L(s) = 1 | − 1.35·3-s + 0.415·7-s − 1.15·9-s + 5.79·11-s + 2.49·13-s + 2.59·17-s + 5.12·19-s − 0.563·21-s − 6.53·23-s + 5.64·27-s + 29-s − 2.03·31-s − 7.85·33-s − 5.31·37-s − 3.38·39-s − 2.86·41-s − 6.71·43-s − 11.7·47-s − 6.82·49-s − 3.52·51-s + 13.2·53-s − 6.94·57-s + 12.1·59-s + 11.6·61-s − 0.481·63-s + 10.0·67-s + 8.86·69-s + ⋯ |
| L(s) = 1 | − 0.783·3-s + 0.156·7-s − 0.386·9-s + 1.74·11-s + 0.692·13-s + 0.629·17-s + 1.17·19-s − 0.123·21-s − 1.36·23-s + 1.08·27-s + 0.185·29-s − 0.365·31-s − 1.36·33-s − 0.873·37-s − 0.542·39-s − 0.447·41-s − 1.02·43-s − 1.70·47-s − 0.975·49-s − 0.493·51-s + 1.81·53-s − 0.920·57-s + 1.58·59-s + 1.49·61-s − 0.0606·63-s + 1.22·67-s + 1.06·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533919433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533919433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 - 0.415T + 7T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 6.53T + 23T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 + 6.71T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 3.64T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−8.600179495372489544657761827263, −8.222231188410914067317226513025, −6.95725044865998407650047467072, −6.52240741524151069957479444073, −5.66078172527984978568651891932, −5.11736191886857303299865453541, −3.90749594909225304479733580314, −3.37668493991272154366820849930, −1.82352360344529469653846637543, −0.832453636586300889654218911312,
0.832453636586300889654218911312, 1.82352360344529469653846637543, 3.37668493991272154366820849930, 3.90749594909225304479733580314, 5.11736191886857303299865453541, 5.66078172527984978568651891932, 6.52240741524151069957479444073, 6.95725044865998407650047467072, 8.222231188410914067317226513025, 8.600179495372489544657761827263
