| L(s) = 1 | − 2-s − 0.533·3-s + 4-s − 2.79·5-s + 0.533·6-s + 0.904·7-s − 8-s − 2.71·9-s + 2.79·10-s + 2.41·11-s − 0.533·12-s − 0.904·14-s + 1.49·15-s + 16-s + 0.336·17-s + 2.71·18-s + 19-s − 2.79·20-s − 0.483·21-s − 2.41·22-s − 8.62·23-s + 0.533·24-s + 2.82·25-s + 3.05·27-s + 0.904·28-s + 7.46·29-s − 1.49·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.308·3-s + 0.5·4-s − 1.25·5-s + 0.218·6-s + 0.342·7-s − 0.353·8-s − 0.904·9-s + 0.884·10-s + 0.727·11-s − 0.154·12-s − 0.241·14-s + 0.385·15-s + 0.250·16-s + 0.0816·17-s + 0.639·18-s + 0.229·19-s − 0.625·20-s − 0.105·21-s − 0.514·22-s − 1.79·23-s + 0.109·24-s + 0.564·25-s + 0.587·27-s + 0.171·28-s + 1.38·29-s − 0.272·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.533T + 3T^{2} \) |
| 5 | \( 1 + 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.904T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 17 | \( 1 - 0.336T + 17T^{2} \) |
| 23 | \( 1 + 8.62T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 3.95T + 47T^{2} \) |
| 53 | \( 1 - 6.74T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−7.957790988173240387679020874258, −6.98650118479935300556207952924, −6.46694811189649571736142833509, −5.64572508941370653948868778049, −4.75196618512380945604408021983, −3.93750252008222419122168833937, −3.22603230519642183445576226131, −2.20354174915350718424224165098, −0.991973080361210316215755650144, 0,
0.991973080361210316215755650144, 2.20354174915350718424224165098, 3.22603230519642183445576226131, 3.93750252008222419122168833937, 4.75196618512380945604408021983, 5.64572508941370653948868778049, 6.46694811189649571736142833509, 6.98650118479935300556207952924, 7.957790988173240387679020874258
