| L(s) = 1 | − 2-s + 2.35·3-s + 4-s + 5-s − 2.35·6-s + 2.99·7-s − 8-s + 2.52·9-s − 10-s + 3.68·11-s + 2.35·12-s + 4.94·13-s − 2.99·14-s + 2.35·15-s + 16-s − 3.29·17-s − 2.52·18-s − 5.50·19-s + 20-s + 7.04·21-s − 3.68·22-s + 9.13·23-s − 2.35·24-s + 25-s − 4.94·26-s − 1.11·27-s + 2.99·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s + 0.447·5-s − 0.959·6-s + 1.13·7-s − 0.353·8-s + 0.842·9-s − 0.316·10-s + 1.11·11-s + 0.678·12-s + 1.37·13-s − 0.801·14-s + 0.607·15-s + 0.250·16-s − 0.799·17-s − 0.595·18-s − 1.26·19-s + 0.223·20-s + 1.53·21-s − 0.785·22-s + 1.90·23-s − 0.479·24-s + 0.200·25-s − 0.970·26-s − 0.213·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.240189068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.240189068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 + 5.50T + 19T^{2} \) |
| 23 | \( 1 - 9.13T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 + 6.71T + 31T^{2} \) |
| 37 | \( 1 - 8.15T + 37T^{2} \) |
| 41 | \( 1 + 3.33T + 41T^{2} \) |
| 43 | \( 1 - 5.63T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 59 | \( 1 + 2.85T + 59T^{2} \) |
| 61 | \( 1 - 5.94T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 - 4.25T + 71T^{2} \) |
| 73 | \( 1 + 3.45T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 + 7.84T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.543900336877339044557153642142, −8.056429168649472757196200951939, −7.14122157825673631035271550308, −6.50319214371259166150871590850, −5.60352533571156885995141012750, −4.40652277946641405028883600514, −3.75015235285469263294209470013, −2.71413247526672568483657427377, −1.86221032984060225220175270108, −1.22611323990696017491648358568,
1.22611323990696017491648358568, 1.86221032984060225220175270108, 2.71413247526672568483657427377, 3.75015235285469263294209470013, 4.40652277946641405028883600514, 5.60352533571156885995141012750, 6.50319214371259166150871590850, 7.14122157825673631035271550308, 8.056429168649472757196200951939, 8.543900336877339044557153642142
