L(s) = 1 | + 2.56·3-s + 5-s + 0.819·7-s + 3.56·9-s − 5.38·11-s + 2.46·13-s + 2.56·15-s + 4.20·17-s + 5.38·19-s + 2.09·21-s + 23-s + 25-s + 1.43·27-s + 2.35·29-s − 6.66·31-s − 13.7·33-s + 0.819·35-s + 7.42·37-s + 6.31·39-s + 5.64·41-s + 1.90·43-s + 3.56·45-s + 9.33·47-s − 6.32·49-s + 10.7·51-s + 13.8·53-s − 5.38·55-s + ⋯ |
L(s) = 1 | + 1.47·3-s + 0.447·5-s + 0.309·7-s + 1.18·9-s − 1.62·11-s + 0.683·13-s + 0.661·15-s + 1.02·17-s + 1.23·19-s + 0.457·21-s + 0.208·23-s + 0.200·25-s + 0.276·27-s + 0.437·29-s − 1.19·31-s − 2.40·33-s + 0.138·35-s + 1.22·37-s + 1.01·39-s + 0.881·41-s + 0.290·43-s + 0.530·45-s + 1.36·47-s − 0.904·49-s + 1.50·51-s + 1.90·53-s − 0.726·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.303154066\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.303154066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 0.819T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 - 1.90T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 5.27T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 3.32T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 + 9.02T + 97T^{2} \) |
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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
−9.175437949302342611283303283211, −8.460628203013040514952497019110, −7.66033047081317460720514849764, −7.40792070593035994203281638614, −5.86517684783595147408429558377, −5.27432250908996983794137005941, −4.07566036678389405489124712426, −3.05302530070692717602824717491, −2.53184868439352273767761380699, −1.29240558217508469784683339977,
1.29240558217508469784683339977, 2.53184868439352273767761380699, 3.05302530070692717602824717491, 4.07566036678389405489124712426, 5.27432250908996983794137005941, 5.86517684783595147408429558377, 7.40792070593035994203281638614, 7.66033047081317460720514849764, 8.460628203013040514952497019110, 9.175437949302342611283303283211