| L(s) = 1 | + 3.32i·3-s − 1.32i·7-s − 8.02·9-s + 5.32·11-s − 5.02i·13-s + 6.34i·17-s + 4.34·19-s + 4.38·21-s − 1.70i·23-s − 16.6i·27-s + 29-s + 8.34·31-s + 17.6i·33-s + 6.93i·37-s + 16.6·39-s + ⋯ |
| L(s) = 1 | + 1.91i·3-s − 0.499i·7-s − 2.67·9-s + 1.60·11-s − 1.39i·13-s + 1.53i·17-s + 0.997·19-s + 0.957·21-s − 0.356i·23-s − 3.21i·27-s + 0.185·29-s + 1.49·31-s + 3.07i·33-s + 1.14i·37-s + 2.67·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.907991127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.907991127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.32iT - 3T^{2} \) |
| 7 | \( 1 + 1.32iT - 7T^{2} \) |
| 11 | \( 1 - 5.32T + 11T^{2} \) |
| 13 | \( 1 + 5.02iT - 13T^{2} \) |
| 17 | \( 1 - 6.34iT - 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 1.70iT - 23T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 6.93iT - 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 - 10.7iT - 43T^{2} \) |
| 47 | \( 1 - 0.679iT - 47T^{2} \) |
| 53 | \( 1 - 2.38iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 - 5.70iT - 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 + 6.73iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.96iT - 83T^{2} \) |
| 89 | \( 1 + 2.58T + 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−9.168808684696348936758518713465, −8.419290309315023821305993803093, −7.81726887498035459006856975824, −6.35892965880746102328801111308, −5.95006218842075434436456532676, −4.85959900155327462509477696747, −4.31737817361596095833677590014, −3.51865179567782703564229555589, −2.97110453432116412779188712728, −1.08786333741248214862620771551,
0.74714789826024250908497805750, 1.66284136756615919445624543681, 2.46695073368184012527957424173, 3.47286901084952499372442918640, 4.75525054152828781931420743352, 5.78609873738579538250312131314, 6.42316289941697772913270207705, 7.10566472384534556754457525087, 7.42039572754321893419614562833, 8.560198263888960675763607158861
