| L(s) = 1 | + 2·2-s + 8.79·3-s + 4·4-s + 17.5·6-s − 24.8·7-s + 8·8-s + 50.3·9-s + 1.58·11-s + 35.1·12-s + 68.9·13-s − 49.6·14-s + 16·16-s + 17·17-s + 100.·18-s + 52.5·19-s − 218.·21-s + 3.16·22-s + 154.·23-s + 70.3·24-s + 137.·26-s + 205.·27-s − 99.3·28-s − 9.87·29-s + 133.·31-s + 32·32-s + 13.9·33-s + 34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s + 1.19·6-s − 1.34·7-s + 0.353·8-s + 1.86·9-s + 0.0433·11-s + 0.846·12-s + 1.47·13-s − 0.947·14-s + 0.250·16-s + 0.242·17-s + 1.31·18-s + 0.634·19-s − 2.26·21-s + 0.0306·22-s + 1.39·23-s + 0.598·24-s + 1.04·26-s + 1.46·27-s − 0.670·28-s − 0.0632·29-s + 0.776·31-s + 0.176·32-s + 0.0733·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.012965311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.012965311\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 8.79T + 27T^{2} \) |
| 7 | \( 1 + 24.8T + 343T^{2} \) |
| 11 | \( 1 - 1.58T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.9T + 2.19e3T^{2} \) |
| 19 | \( 1 - 52.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 9.87T + 2.43e4T^{2} \) |
| 31 | \( 1 - 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 16.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 224.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 302.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 469.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 776.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 74.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 418.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.36e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 625.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 79.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 537.T + 9.12e5T^{2} \) |
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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
−9.678654711447216254922855025663, −8.887948044891874247148878775280, −8.241981681474551466423771594513, −7.10250905508710578069402497897, −6.53679102325318410285969411960, −5.30552456524428156601753789239, −3.85256442497004435131546804531, −3.39283758541289032204339456888, −2.64529183326153384483534605177, −1.25301983760884596876894601273,
1.25301983760884596876894601273, 2.64529183326153384483534605177, 3.39283758541289032204339456888, 3.85256442497004435131546804531, 5.30552456524428156601753789239, 6.53679102325318410285969411960, 7.10250905508710578069402497897, 8.241981681474551466423771594513, 8.887948044891874247148878775280, 9.678654711447216254922855025663
