| L(s) = 1 | + 1.05·2-s − 3-s − 0.894·4-s − 4.26·5-s − 1.05·6-s − 3.04·8-s + 9-s − 4.48·10-s − 2.24·11-s + 0.894·12-s + 0.346·13-s + 4.26·15-s − 1.41·16-s + 4.66·17-s + 1.05·18-s − 2.00·19-s + 3.81·20-s − 2.36·22-s + 2.47·23-s + 3.04·24-s + 13.1·25-s + 0.363·26-s − 27-s + 6.18·29-s + 4.48·30-s − 3.71·31-s + 4.60·32-s + ⋯ |
| L(s) = 1 | + 0.743·2-s − 0.577·3-s − 0.447·4-s − 1.90·5-s − 0.429·6-s − 1.07·8-s + 0.333·9-s − 1.41·10-s − 0.677·11-s + 0.258·12-s + 0.0959·13-s + 1.10·15-s − 0.352·16-s + 1.13·17-s + 0.247·18-s − 0.458·19-s + 0.852·20-s − 0.503·22-s + 0.515·23-s + 0.621·24-s + 2.63·25-s + 0.0713·26-s − 0.192·27-s + 1.14·29-s + 0.818·30-s − 0.666·31-s + 0.813·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 0.346T + 13T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 + 2.00T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 6.18T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 43 | \( 1 + 0.374T + 43T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 7.37T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.77269164452893109031338843873, −6.97864434430487417761437068578, −6.24791426284006515419368563467, −5.12076205458156100669386407196, −4.99460696298049596237653175328, −3.99707233288993233765335586864, −3.56952616345526241013420810766, −2.76652190482375201280295215543, −0.920064999201103989800221179637, 0,
0.920064999201103989800221179637, 2.76652190482375201280295215543, 3.56952616345526241013420810766, 3.99707233288993233765335586864, 4.99460696298049596237653175328, 5.12076205458156100669386407196, 6.24791426284006515419368563467, 6.97864434430487417761437068578, 7.77269164452893109031338843873
