| L(s) = 1 | + 1.41·5-s + 4·11-s + 1.41·13-s − 4·19-s − 5.65·23-s − 2.99·25-s − 7.07·29-s + 5.65·31-s − 4.24·37-s − 12·43-s + 11.3·47-s − 7·49-s + 1.41·53-s + 5.65·55-s − 4·59-s − 12.7·61-s + 2.00·65-s − 4·67-s − 5.65·71-s − 10·73-s − 16.9·79-s + 12·83-s − 6·89-s − 5.65·95-s + 8·97-s − 1.41·101-s + 12·107-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 1.20·11-s + 0.392·13-s − 0.917·19-s − 1.17·23-s − 0.599·25-s − 1.31·29-s + 1.01·31-s − 0.697·37-s − 1.82·43-s + 1.65·47-s − 49-s + 0.194·53-s + 0.762·55-s − 0.520·59-s − 1.62·61-s + 0.248·65-s − 0.488·67-s − 0.671·71-s − 1.17·73-s − 1.90·79-s + 1.31·83-s − 0.635·89-s − 0.580·95-s + 0.812·97-s − 0.140·101-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.36612501016609359836750826425, −6.45206427923396380051247402916, −6.16560180748490936870027295396, −5.46852623312277093154441666002, −4.43619876001694472641317193489, −3.93732789454797866047597312211, −3.07926334385005403221238299790, −1.94811419374731618089291410793, −1.49487605456622995793581833708, 0,
1.49487605456622995793581833708, 1.94811419374731618089291410793, 3.07926334385005403221238299790, 3.93732789454797866047597312211, 4.43619876001694472641317193489, 5.46852623312277093154441666002, 6.16560180748490936870027295396, 6.45206427923396380051247402916, 7.36612501016609359836750826425
