| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 33-s + 35-s + 37-s + 39-s − 41-s + 43-s − 45-s + 47-s + 49-s + 51-s − 53-s + 55-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 33-s + 35-s + 37-s + 39-s − 41-s + 43-s − 45-s + 47-s + 49-s + 51-s − 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.569590663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569590663\) |
| \(L(1)\) |
\(\approx\) |
\(1.209973622\) |
| \(L(1)\) |
\(\approx\) |
\(1.209973622\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.6534526552990302421273466938, −21.73731701155006765257950825007, −20.66628910129024682985481211719, −20.2179294981673142869390581946, −19.37071792898361337530382239141, −18.6727185709864360469897051694, −18.14067684053386849683483934057, −16.353622416277015320606286681623, −15.99467614669736435864163637486, −15.39258273840824963345088935123, −14.32584640276196965520105463794, −13.52475945185674948898715694970, −12.72405840563183766196027857949, −11.99018832969722033231169154956, −10.73339454674448532242113531299, −9.947081161943796718064811096379, −9.04823579157729795875452715040, −8.042576160241816920559382382656, −7.599459917087642155807653680331, −6.517468965782615046648449493210, −5.29835044227758072206838189739, −3.945698903099543684800690188654, −3.40276785561600293528616283004, −2.55115115369923691397537192188, −0.947188642685941150504459089100,
0.947188642685941150504459089100, 2.55115115369923691397537192188, 3.40276785561600293528616283004, 3.945698903099543684800690188654, 5.29835044227758072206838189739, 6.517468965782615046648449493210, 7.599459917087642155807653680331, 8.042576160241816920559382382656, 9.04823579157729795875452715040, 9.947081161943796718064811096379, 10.73339454674448532242113531299, 11.99018832969722033231169154956, 12.72405840563183766196027857949, 13.52475945185674948898715694970, 14.32584640276196965520105463794, 15.39258273840824963345088935123, 15.99467614669736435864163637486, 16.353622416277015320606286681623, 18.14067684053386849683483934057, 18.6727185709864360469897051694, 19.37071792898361337530382239141, 20.2179294981673142869390581946, 20.66628910129024682985481211719, 21.73731701155006765257950825007, 22.6534526552990302421273466938
