L(s) = 1 | + 3.31i·11-s − 2.79·13-s − 4.88i·17-s + 4.35·19-s − 5·25-s + 7·49-s + 8.72i·53-s + 2.72i·59-s + 14.7i·71-s + 17.1·79-s − 1.75i·83-s − 3.27i·89-s − 13.2i·101-s − 17.4·109-s + 20.7i·113-s + ⋯ |
L(s) = 1 | + 1.00i·11-s − 0.775·13-s − 1.18i·17-s + 1.00·19-s − 25-s + 49-s + 1.19i·53-s + 0.355i·59-s + 1.74i·71-s + 1.92·79-s − 0.192i·83-s − 0.346i·89-s − 1.32i·101-s − 1.67·109-s + 1.94i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.320342450\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320342450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| 19 | \( 1 - 4.35T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 + 4.88iT - 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.72iT - 53T^{2} \) |
| 59 | \( 1 - 2.72iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 14.7iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 1.75iT - 83T^{2} \) |
| 89 | \( 1 + 3.27iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−7.80844506687874279226971656608, −7.38390338720132679826217221268, −6.88966036490446343580110387190, −5.88642044267051179172700670178, −5.18871050149975344473231467998, −4.60415017282693730568558099230, −3.78729339707179229432266287478, −2.78408458280605220181841940106, −2.14641450627228403128946268609, −0.975413134280614393802005143722,
0.35544010580295572867287398026, 1.54080482326575401819105250747, 2.48852958374385687561246549992, 3.42695994872383856084380387440, 3.97867928390240849805444290673, 5.01878147752781845487843716308, 5.61217211956235493564289530179, 6.28212979488752551640863389536, 7.00528335766621916188209368420, 7.932357598514393710540418701146