| L(s) = 1 | + 4.79·7-s + 11-s + 13-s − 3.79·17-s − 2.58·19-s + 0.791·23-s − 2.20·29-s + 0.582·31-s − 6.58·37-s + 10.5·41-s + 10·43-s − 10.5·47-s + 15.9·49-s − 2.37·53-s + 1.41·59-s + 8.79·61-s + 4·67-s − 16.7·71-s + 3.20·73-s + 4.79·77-s + 16.5·79-s + 12.9·83-s − 3.79·89-s + 4.79·91-s + 10.7·97-s + 3.62·101-s + 16.9·103-s + ⋯ |
| L(s) = 1 | + 1.81·7-s + 0.301·11-s + 0.277·13-s − 0.919·17-s − 0.592·19-s + 0.164·23-s − 0.410·29-s + 0.104·31-s − 1.08·37-s + 1.65·41-s + 1.52·43-s − 1.54·47-s + 2.27·49-s − 0.326·53-s + 0.184·59-s + 1.12·61-s + 0.488·67-s − 1.98·71-s + 0.375·73-s + 0.546·77-s + 1.86·79-s + 1.42·83-s − 0.401·89-s + 0.502·91-s + 1.09·97-s + 0.360·101-s + 1.67·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.727956603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.727956603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.79T + 7T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 0.791T + 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 - 3.20T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.72912935320488741576081858349, −7.08107739704311885943135516497, −6.28133678762413112518611385905, −5.57033630278259973078858514524, −4.77026291317243699053346789736, −4.37064541850107564091553548413, −3.54767756146249971842748229656, −2.31932786640109033913418928665, −1.81330415912202656587947769490, −0.807390677757020341734658265934,
0.807390677757020341734658265934, 1.81330415912202656587947769490, 2.31932786640109033913418928665, 3.54767756146249971842748229656, 4.37064541850107564091553548413, 4.77026291317243699053346789736, 5.57033630278259973078858514524, 6.28133678762413112518611385905, 7.08107739704311885943135516497, 7.72912935320488741576081858349
