L(s) = 1 | + 1.05·2-s − 0.881·4-s − 1.01·7-s − 3.04·8-s − 5.12·11-s − 6.08·13-s − 1.07·14-s − 1.45·16-s + 3.19·17-s + 3.42·19-s − 5.41·22-s + 2.91·23-s − 6.43·26-s + 0.892·28-s + 1.55·29-s − 7.99·31-s + 4.55·32-s + 3.37·34-s − 8.40·37-s + 3.62·38-s + 1.86·41-s + 5.22·43-s + 4.51·44-s + 3.08·46-s − 4.80·47-s − 5.97·49-s + 5.36·52-s + ⋯ |
L(s) = 1 | + 0.747·2-s − 0.440·4-s − 0.382·7-s − 1.07·8-s − 1.54·11-s − 1.68·13-s − 0.285·14-s − 0.364·16-s + 0.774·17-s + 0.786·19-s − 1.15·22-s + 0.608·23-s − 1.26·26-s + 0.168·28-s + 0.288·29-s − 1.43·31-s + 0.804·32-s + 0.579·34-s − 1.38·37-s + 0.588·38-s + 0.291·41-s + 0.796·43-s + 0.680·44-s + 0.455·46-s − 0.700·47-s − 0.853·49-s + 0.744·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.166678620\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.166678620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 6.08T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 + 4.64T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 0.595T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 6.39T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.88090065946570510040445876260, −7.52596522832419635077353672990, −6.67141259074697931967860856180, −5.59473350815689075294772061774, −5.19728649911297995676712963861, −4.75008974456326732828025993946, −3.57104692438167243355341813433, −3.02504323661848858512291803553, −2.20855647800228073338567591254, −0.48534779452236361034386564443,
0.48534779452236361034386564443, 2.20855647800228073338567591254, 3.02504323661848858512291803553, 3.57104692438167243355341813433, 4.75008974456326732828025993946, 5.19728649911297995676712963861, 5.59473350815689075294772061774, 6.67141259074697931967860856180, 7.52596522832419635077353672990, 7.88090065946570510040445876260