L(s) = 1 | + 3.55·2-s − 3.90·3-s + 4.67·4-s + 19.3·5-s − 13.8·6-s − 18.4·7-s − 11.8·8-s − 11.7·9-s + 68.8·10-s + 11·11-s − 18.2·12-s − 65.6·14-s − 75.4·15-s − 79.5·16-s − 18.2·17-s − 41.9·18-s + 130.·19-s + 90.3·20-s + 71.8·21-s + 39.1·22-s + 69.1·23-s + 46.2·24-s + 248.·25-s + 151.·27-s − 86.1·28-s − 92.1·29-s − 268.·30-s + ⋯ |
L(s) = 1 | + 1.25·2-s − 0.750·3-s + 0.584·4-s + 1.72·5-s − 0.944·6-s − 0.995·7-s − 0.523·8-s − 0.436·9-s + 2.17·10-s + 0.301·11-s − 0.438·12-s − 1.25·14-s − 1.29·15-s − 1.24·16-s − 0.260·17-s − 0.549·18-s + 1.57·19-s + 1.01·20-s + 0.747·21-s + 0.379·22-s + 0.627·23-s + 0.392·24-s + 1.99·25-s + 1.07·27-s − 0.581·28-s − 0.590·29-s − 1.63·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 3.55T + 8T^{2} \) |
| 3 | \( 1 + 3.90T + 27T^{2} \) |
| 5 | \( 1 - 19.3T + 125T^{2} \) |
| 7 | \( 1 + 18.4T + 343T^{2} \) |
| 17 | \( 1 + 18.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 92.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.39T + 6.89e4T^{2} \) |
| 43 | \( 1 + 549.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 237.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 179.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 805.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 134.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 555.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 202.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 436.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.16e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 684.T + 9.12e5T^{2} \) |
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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
−8.832490676698047982336406628721, −7.15658853471620850532714795357, −6.41509523128351519100054498430, −5.86942252147936617928642998417, −5.42746713072486926731015564997, −4.68101028841274510867947033778, −3.29922315361003862323216789117, −2.79206206080274800772212878451, −1.46631049223698745985483358303, 0,
1.46631049223698745985483358303, 2.79206206080274800772212878451, 3.29922315361003862323216789117, 4.68101028841274510867947033778, 5.42746713072486926731015564997, 5.86942252147936617928642998417, 6.41509523128351519100054498430, 7.15658853471620850532714795357, 8.832490676698047982336406628721