| L(s) = 1 | + 0.414·3-s − 3.82·5-s + 1.58·7-s − 2.82·9-s − 4.82·11-s − 13-s − 1.58·15-s + 17-s − 5.65·19-s + 0.656·21-s + 3.17·23-s + 9.65·25-s − 2.41·27-s + 7.65·29-s − 7.65·31-s − 1.99·33-s − 6.07·35-s − 7·37-s − 0.414·39-s − 1.65·41-s + 5.58·43-s + 10.8·45-s + 9.24·47-s − 4.48·49-s + 0.414·51-s + 7.65·53-s + 18.4·55-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 1.71·5-s + 0.599·7-s − 0.942·9-s − 1.45·11-s − 0.277·13-s − 0.409·15-s + 0.242·17-s − 1.29·19-s + 0.143·21-s + 0.661·23-s + 1.93·25-s − 0.464·27-s + 1.42·29-s − 1.37·31-s − 0.348·33-s − 1.02·35-s − 1.15·37-s − 0.0663·39-s − 0.258·41-s + 0.851·43-s + 1.61·45-s + 1.34·47-s − 0.640·49-s + 0.0580·51-s + 1.05·53-s + 2.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7461205409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7461205409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 2.34T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.610821912189475961278961746741, −7.76274096674665834953298325581, −7.56744055664395896508286018276, −6.51070934925789316025165044677, −5.33776552483667442058432766992, −4.81457128932651728844275146505, −3.89886203516659175225079807684, −3.08122411075702329249883067410, −2.26372996539020685859188889576, −0.47999263220662198682048470789,
0.47999263220662198682048470789, 2.26372996539020685859188889576, 3.08122411075702329249883067410, 3.89886203516659175225079807684, 4.81457128932651728844275146505, 5.33776552483667442058432766992, 6.51070934925789316025165044677, 7.56744055664395896508286018276, 7.76274096674665834953298325581, 8.610821912189475961278961746741
