L(s) = 1 | − 2.56i·3-s − 3.59·9-s − 1.56·11-s + 5.56i·13-s − 7.16i·17-s − 3.16·19-s + 5.73i·23-s + 1.53i·27-s − 1.96·29-s + 0.969·31-s + 4.03i·33-s + 6.70i·37-s + 14.3·39-s − 8.87·41-s − 4.59i·43-s + ⋯ |
L(s) = 1 | − 1.48i·3-s − 1.19·9-s − 0.473·11-s + 1.54i·13-s − 1.73i·17-s − 0.726·19-s + 1.19i·23-s + 0.296i·27-s − 0.365·29-s + 0.174·31-s + 0.701i·33-s + 1.10i·37-s + 2.29·39-s − 1.38·41-s − 0.701i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.012431519\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.012431519\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.56iT - 3T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 5.56iT - 13T^{2} \) |
| 17 | \( 1 + 7.16iT - 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - 5.73iT - 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 - 0.969T + 31T^{2} \) |
| 37 | \( 1 - 6.70iT - 37T^{2} \) |
| 41 | \( 1 + 8.87T + 41T^{2} \) |
| 43 | \( 1 + 4.59iT - 43T^{2} \) |
| 47 | \( 1 + 0.401iT - 47T^{2} \) |
| 53 | \( 1 - 9.53iT - 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 0.862iT - 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 17.1iT - 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 + 0.233iT - 97T^{2} \) |
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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
−8.150706955671061511502120096245, −7.43354570037619734252315959820, −6.91006380041373922711757139874, −6.47551595008260381313012723742, −5.47622188401720776042382106008, −4.77064949861609656439751555181, −3.70324036440147196585265443148, −2.58996723990306615624932240119, −1.97036874748756152739925885490, −1.00853614661987004619422680489,
0.29937643067960263923899077701, 2.01206737868964540549006109155, 3.08486663709978039656741118126, 3.74397713929177276596976457486, 4.44451170482676578153081543929, 5.22778297139719097731349798604, 5.80586042224349870677538136849, 6.59534947559320869147829653902, 7.73658875074243932112460892729, 8.455187308083686873137630168526