| L(s) = 1 | + 2-s + 2.48·3-s + 4-s + 1.48·5-s + 2.48·6-s + 4.90·7-s + 8-s + 3.18·9-s + 1.48·10-s + 2.48·12-s − 3.09·13-s + 4.90·14-s + 3.70·15-s + 16-s − 5.90·17-s + 3.18·18-s + 19-s + 1.48·20-s + 12.1·21-s + 2.09·23-s + 2.48·24-s − 2.78·25-s − 3.09·26-s + 0.468·27-s + 4.90·28-s − 0.700·29-s + 3.70·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.43·3-s + 0.5·4-s + 0.665·5-s + 1.01·6-s + 1.85·7-s + 0.353·8-s + 1.06·9-s + 0.470·10-s + 0.718·12-s − 0.857·13-s + 1.31·14-s + 0.955·15-s + 0.250·16-s − 1.43·17-s + 0.751·18-s + 0.229·19-s + 0.332·20-s + 2.66·21-s + 0.436·23-s + 0.507·24-s − 0.557·25-s − 0.606·26-s + 0.0900·27-s + 0.926·28-s − 0.130·29-s + 0.675·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.941703207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.941703207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 - 4.90T + 7T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 23 | \( 1 - 2.09T + 23T^{2} \) |
| 29 | \( 1 + 0.700T + 29T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 - 9.31T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 0.792T + 43T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 0.603T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.178396162297631613157856639794, −7.76751668048463501075528547748, −7.00852046259641683365981854813, −6.07896903445151958285432444669, −5.04373487579619737031195554456, −4.64066644005231217420585661219, −3.83660789915673565219901422352, −2.63201622760380838020872472529, −2.23598363123536398039563528308, −1.46626578204086222463185624527,
1.46626578204086222463185624527, 2.23598363123536398039563528308, 2.63201622760380838020872472529, 3.83660789915673565219901422352, 4.64066644005231217420585661219, 5.04373487579619737031195554456, 6.07896903445151958285432444669, 7.00852046259641683365981854813, 7.76751668048463501075528547748, 8.178396162297631613157856639794
