| L(s) = 1 | − 2-s − 1.54·3-s + 4-s + 1.89·5-s + 1.54·6-s − 0.915·7-s − 8-s − 0.611·9-s − 1.89·10-s − 4.21·11-s − 1.54·12-s + 0.915·14-s − 2.93·15-s + 16-s − 4.17·17-s + 0.611·18-s + 19-s + 1.89·20-s + 1.41·21-s + 4.21·22-s − 3.15·23-s + 1.54·24-s − 1.39·25-s + 5.58·27-s − 0.915·28-s + 8.44·29-s + 2.93·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.892·3-s + 0.5·4-s + 0.849·5-s + 0.630·6-s − 0.346·7-s − 0.353·8-s − 0.203·9-s − 0.600·10-s − 1.27·11-s − 0.446·12-s + 0.244·14-s − 0.757·15-s + 0.250·16-s − 1.01·17-s + 0.144·18-s + 0.229·19-s + 0.424·20-s + 0.308·21-s + 0.899·22-s − 0.658·23-s + 0.315·24-s − 0.278·25-s + 1.07·27-s − 0.173·28-s + 1.56·29-s + 0.535·30-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 | \( 1 + 1.54T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 7 | \( 1 + 0.915T + 7T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 23 | \( 1 + 3.15T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 5.54T + 31T^{2} \) |
| 37 | \( 1 - 0.550T + 37T^{2} \) |
| 41 | \( 1 + 0.115T + 41T^{2} \) |
| 43 | \( 1 - 13.0T + 43T^{2} \) |
| 47 | \( 1 - 5.54T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 5.36T + 59T^{2} \) |
| 61 | \( 1 - 2.80T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 0.0498T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 4.06T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.75815112129223034327717289569, −6.78691078402615055406755446006, −6.28775437244870408635366651410, −5.67710630322425289389987262795, −5.08991490337558429109972406318, −4.14015330616075962473136765870, −2.67253874003730174448569904476, −2.43087625141229177327485824089, −1.01565741679024371558460375066, 0,
1.01565741679024371558460375066, 2.43087625141229177327485824089, 2.67253874003730174448569904476, 4.14015330616075962473136765870, 5.08991490337558429109972406318, 5.67710630322425289389987262795, 6.28775437244870408635366651410, 6.78691078402615055406755446006, 7.75815112129223034327717289569
