| L(s) = 1 | + 5-s − 3.27·7-s + 6.27·11-s − 1.27·13-s + 2·17-s + 19-s − 7.27·23-s + 25-s + 6.27·29-s + 6.27·31-s − 3.27·35-s − 10.5·37-s + 7.54·41-s + 4·43-s − 1.27·47-s + 3.72·49-s + 0.725·53-s + 6.27·55-s − 13·59-s + 8.54·61-s − 1.27·65-s + 0.549·67-s − 8.27·71-s + 15.0·73-s − 20.5·77-s + 10.5·79-s − 2.54·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.23·7-s + 1.89·11-s − 0.353·13-s + 0.485·17-s + 0.229·19-s − 1.51·23-s + 0.200·25-s + 1.16·29-s + 1.12·31-s − 0.553·35-s − 1.73·37-s + 1.17·41-s + 0.609·43-s − 0.185·47-s + 0.532·49-s + 0.0995·53-s + 0.846·55-s − 1.69·59-s + 1.09·61-s − 0.158·65-s + 0.0671·67-s − 0.982·71-s + 1.76·73-s − 2.34·77-s + 1.18·79-s − 0.279·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.908939684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.908939684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.27T + 7T^{2} \) |
| 11 | \( 1 - 6.27T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 7.54T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 - 0.725T + 53T^{2} \) |
| 59 | \( 1 + 13T + 59T^{2} \) |
| 61 | \( 1 - 8.54T + 61T^{2} \) |
| 67 | \( 1 - 0.549T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 2.54T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−8.857449078695155521324121511884, −7.917225906548906442004130191206, −6.93137968145181140621046171731, −6.35870535421854895138558621178, −5.93421836585419308708142466618, −4.73688837916018369756851686823, −3.84430790532800964179544769989, −3.16432154335307708493304060967, −2.03295347094569248608226426374, −0.844460583937210432502758867826,
0.844460583937210432502758867826, 2.03295347094569248608226426374, 3.16432154335307708493304060967, 3.84430790532800964179544769989, 4.73688837916018369756851686823, 5.93421836585419308708142466618, 6.35870535421854895138558621178, 6.93137968145181140621046171731, 7.917225906548906442004130191206, 8.857449078695155521324121511884
