| L(s) = 1 | + 1.48i·3-s + 1.02·5-s + (1.70 + 2.02i)7-s + 0.784·9-s − 0.608·11-s − 13-s + 1.53i·15-s + 0.526i·17-s + 1.94i·19-s + (−3.00 + 2.54i)21-s − 0.0153i·23-s − 3.93·25-s + 5.63i·27-s + 4.64i·29-s − 4.57·31-s + ⋯ |
| L(s) = 1 | + 0.859i·3-s + 0.460·5-s + (0.645 + 0.763i)7-s + 0.261·9-s − 0.183·11-s − 0.277·13-s + 0.395i·15-s + 0.127i·17-s + 0.446i·19-s + (−0.656 + 0.554i)21-s − 0.00319i·23-s − 0.787·25-s + 1.08i·27-s + 0.862i·29-s − 0.821·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.929326878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.929326878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.70 - 2.02i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.48iT - 3T^{2} \) |
| 5 | \( 1 - 1.02T + 5T^{2} \) |
| 11 | \( 1 + 0.608T + 11T^{2} \) |
| 17 | \( 1 - 0.526iT - 17T^{2} \) |
| 19 | \( 1 - 1.94iT - 19T^{2} \) |
| 23 | \( 1 + 0.0153iT - 23T^{2} \) |
| 29 | \( 1 - 4.64iT - 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 + 2.15iT - 37T^{2} \) |
| 41 | \( 1 - 6.24iT - 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 1.16iT - 53T^{2} \) |
| 59 | \( 1 - 0.783iT - 59T^{2} \) |
| 61 | \( 1 - 5.32T + 61T^{2} \) |
| 67 | \( 1 - 4.18T + 67T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 - 9.18iT - 73T^{2} \) |
| 79 | \( 1 - 9.29iT - 79T^{2} \) |
| 83 | \( 1 - 1.45iT - 83T^{2} \) |
| 89 | \( 1 - 12.0iT - 89T^{2} \) |
| 97 | \( 1 - 8.46iT - 97T^{2} \) |
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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
−9.265653042876221432270738307644, −8.352343854195057029027565978546, −7.64939568746445699049065649183, −6.71382235039557909659357382286, −5.67824795986575103284917241484, −5.23435725746594349280335231034, −4.38695385655357670035262744298, −3.55475893414556212043393166682, −2.42019722376618314655498350663, −1.51129202551398363131461236545,
0.60270205181872971312496994229, 1.70444000751775205031346427497, 2.41579444571388446737010271363, 3.78833086860576249488293118381, 4.58436089083833889934419581951, 5.49561107688205573639043161066, 6.31814449966417773989287142389, 7.19503094514381079236296920395, 7.54101973443787636478039912774, 8.308755404185861477451730108528
