| L(s) = 1 | − 1.34·3-s − 1.18·9-s + 1.41·11-s + 6.47·13-s − 2.59·17-s + 1.16·19-s + 1.57·23-s + 5.63·27-s + 6.22·29-s + 4.98·31-s − 1.90·33-s + 8.47·37-s − 8.72·39-s + 3.57·41-s + 7.24·43-s − 10.9·47-s + 3.49·51-s − 12.7·53-s − 1.56·57-s − 2.04·59-s − 0.615·61-s − 6.08·67-s − 2.12·69-s − 8.27·71-s − 3.51·73-s + 0.879·79-s − 4.04·81-s + ⋯ |
| L(s) = 1 | − 0.777·3-s − 0.394·9-s + 0.425·11-s + 1.79·13-s − 0.629·17-s + 0.266·19-s + 0.328·23-s + 1.08·27-s + 1.15·29-s + 0.895·31-s − 0.331·33-s + 1.39·37-s − 1.39·39-s + 0.558·41-s + 1.10·43-s − 1.59·47-s + 0.489·51-s − 1.75·53-s − 0.207·57-s − 0.265·59-s − 0.0788·61-s − 0.743·67-s − 0.255·69-s − 0.982·71-s − 0.411·73-s + 0.0989·79-s − 0.449·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684229754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684229754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 - 1.57T + 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 - 7.24T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 2.04T + 59T^{2} \) |
| 61 | \( 1 + 0.615T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 + 3.51T + 73T^{2} \) |
| 79 | \( 1 - 0.879T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 - 5.96T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−7.73018506358386274586156848451, −6.68320821737687771447340154102, −6.18146801844075539295611111770, −5.93917564038816579709443903226, −4.83200045313096764305124246077, −4.38658836072135498675025077670, −3.39003110119113120804428162268, −2.72093333382598850396946269151, −1.44164118828425535609646011443, −0.70232850301197657098232438321,
0.70232850301197657098232438321, 1.44164118828425535609646011443, 2.72093333382598850396946269151, 3.39003110119113120804428162268, 4.38658836072135498675025077670, 4.83200045313096764305124246077, 5.93917564038816579709443903226, 6.18146801844075539295611111770, 6.68320821737687771447340154102, 7.73018506358386274586156848451
