| L(s) = 1 | + 1.62·3-s + 3.78·5-s − 2.93·7-s − 0.356·9-s − 5.63·11-s + 2.51·13-s + 6.15·15-s − 2.13·17-s − 5.04·19-s − 4.77·21-s − 5.96·23-s + 9.35·25-s − 5.45·27-s − 5.08·29-s − 5.14·31-s − 9.16·33-s − 11.1·35-s + 6.06·37-s + 4.09·39-s + 1.12·41-s + 11.2·43-s − 1.35·45-s − 7.62·47-s + 1.63·49-s − 3.46·51-s − 9.20·53-s − 21.3·55-s + ⋯ |
| L(s) = 1 | + 0.938·3-s + 1.69·5-s − 1.11·7-s − 0.118·9-s − 1.69·11-s + 0.698·13-s + 1.59·15-s − 0.517·17-s − 1.15·19-s − 1.04·21-s − 1.24·23-s + 1.87·25-s − 1.05·27-s − 0.944·29-s − 0.924·31-s − 1.59·33-s − 1.88·35-s + 0.997·37-s + 0.655·39-s + 0.176·41-s + 1.70·43-s − 0.201·45-s − 1.11·47-s + 0.233·49-s − 0.485·51-s − 1.26·53-s − 2.87·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + 5.63T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 + 5.08T + 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 7.62T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 - 2.84T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 + 0.109T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−8.210415501156943662436134631372, −7.46463962845181132731127205584, −6.32876523975476576045084687710, −5.99446867112830800508979406192, −5.32012731254616723150281379223, −4.10281323510241465553071308263, −3.11524731335336132545831804959, −2.42014506053853301132098686969, −1.92696690789202808978909556739, 0,
1.92696690789202808978909556739, 2.42014506053853301132098686969, 3.11524731335336132545831804959, 4.10281323510241465553071308263, 5.32012731254616723150281379223, 5.99446867112830800508979406192, 6.32876523975476576045084687710, 7.46463962845181132731127205584, 8.210415501156943662436134631372
