| L(s) = 1 | + 0.652·5-s − 7-s + 3.41·11-s − 0.305·13-s + 0.226·17-s + 2.16·19-s − 6.70·23-s − 4.57·25-s + 0.509·29-s + 5.71·31-s − 0.652·35-s − 2.28·37-s − 0.958·41-s + 7.71·43-s + 8.29·47-s + 49-s + 3.18·53-s + 2.22·55-s − 12.1·59-s − 3.50·61-s − 0.199·65-s − 5.74·67-s + 14.0·71-s + 12.4·73-s − 3.41·77-s + 11.9·79-s + 4.27·83-s + ⋯ |
| L(s) = 1 | + 0.291·5-s − 0.377·7-s + 1.02·11-s − 0.0847·13-s + 0.0549·17-s + 0.496·19-s − 1.39·23-s − 0.914·25-s + 0.0946·29-s + 1.02·31-s − 0.110·35-s − 0.375·37-s − 0.149·41-s + 1.17·43-s + 1.20·47-s + 0.142·49-s + 0.437·53-s + 0.300·55-s − 1.58·59-s − 0.449·61-s − 0.0247·65-s − 0.701·67-s + 1.66·71-s + 1.45·73-s − 0.388·77-s + 1.34·79-s + 0.468·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.110817959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.110817959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.652T + 5T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 0.305T + 13T^{2} \) |
| 17 | \( 1 - 0.226T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 0.509T + 29T^{2} \) |
| 31 | \( 1 - 5.71T + 31T^{2} \) |
| 37 | \( 1 + 2.28T + 37T^{2} \) |
| 41 | \( 1 + 0.958T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 8.29T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.74198363819550306211886689394, −7.01293988909348912508957273053, −6.20589568053361955392840618346, −5.92677626236596525643964033195, −4.95735880251478022352499483327, −4.09774645952002925146413941287, −3.58656676249718544358578802659, −2.56820976122091824400830153550, −1.76392028998598566113426962756, −0.70854206264898913283623119668,
0.70854206264898913283623119668, 1.76392028998598566113426962756, 2.56820976122091824400830153550, 3.58656676249718544358578802659, 4.09774645952002925146413941287, 4.95735880251478022352499483327, 5.92677626236596525643964033195, 6.20589568053361955392840618346, 7.01293988909348912508957273053, 7.74198363819550306211886689394
