L(s) = 1 | + 2-s + 4-s + 0.193·7-s + 8-s − 0.514·11-s + 3.12·13-s + 0.193·14-s + 16-s + 2.83·17-s + 19-s − 0.514·22-s − 2.32·23-s + 3.12·26-s + 0.193·28-s + 0.164·29-s − 9.05·31-s + 32-s + 2.83·34-s + 3.02·37-s + 38-s + 9.96·41-s + 5.51·43-s − 0.514·44-s − 2.32·46-s − 1.70·47-s − 6.96·49-s + 3.12·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0730·7-s + 0.353·8-s − 0.155·11-s + 0.867·13-s + 0.0516·14-s + 0.250·16-s + 0.687·17-s + 0.229·19-s − 0.109·22-s − 0.483·23-s + 0.613·26-s + 0.0365·28-s + 0.0306·29-s − 1.62·31-s + 0.176·32-s + 0.486·34-s + 0.497·37-s + 0.162·38-s + 1.55·41-s + 0.840·43-s − 0.0775·44-s − 0.342·46-s − 0.249·47-s − 0.994·49-s + 0.433·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.649085658\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.649085658\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 0.193T + 7T^{2} \) |
| 11 | \( 1 + 0.514T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 23 | \( 1 + 2.32T + 23T^{2} \) |
| 29 | \( 1 - 0.164T + 29T^{2} \) |
| 31 | \( 1 + 9.05T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 - 2.90T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 5.92T + 61T^{2} \) |
| 67 | \( 1 - 4.22T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 - 6.02T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 9.01T + 89T^{2} \) |
| 97 | \( 1 + 8.89T + 97T^{2} \) |
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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
−7.70209710193331282617442565880, −7.05888502840594734912265212936, −6.23340956559213953648331700444, −5.66342727451410980038340385076, −5.11214989040029445347142915604, −4.08142468131472553532568346348, −3.67483620174666707327009780563, −2.75639612573205063574328014073, −1.88551383634954147353384526079, −0.858316979468417583863481948420,
0.858316979468417583863481948420, 1.88551383634954147353384526079, 2.75639612573205063574328014073, 3.67483620174666707327009780563, 4.08142468131472553532568346348, 5.11214989040029445347142915604, 5.66342727451410980038340385076, 6.23340956559213953648331700444, 7.05888502840594734912265212936, 7.70209710193331282617442565880