| L(s) = 1 | + i·2-s − 4-s − 1.61i·7-s − i·8-s − 3.38·11-s + 2.61i·13-s + 1.61·14-s + 16-s + 2.47i·17-s + 3.61·19-s − 3.38i·22-s + 0.145i·23-s − 2.61·26-s + 1.61i·28-s + 2.76·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.611i·7-s − 0.353i·8-s − 1.01·11-s + 0.726i·13-s + 0.432·14-s + 0.250·16-s + 0.599i·17-s + 0.830·19-s − 0.721i·22-s + 0.0304i·23-s − 0.513·26-s + 0.305i·28-s + 0.513·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5879920197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5879920197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.61iT - 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 2.61iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 0.145iT - 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 4.38iT - 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 + 1.52iT - 43T^{2} \) |
| 47 | \( 1 - 6.61iT - 47T^{2} \) |
| 53 | \( 1 - 8.56iT - 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.23iT - 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 15.7iT - 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 8.18iT - 97T^{2} \) |
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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
−9.233205790727123330111373289336, −8.610894148969355494332027392606, −7.56579696362243787503735379151, −7.35743612361675351965831552439, −6.29722872547302647543337851168, −5.56238545441893291671446144868, −4.69482832592285289717401600223, −3.91290527951872340157218893484, −2.83916683834031535311387107106, −1.40963399628503020815070486566,
0.20361911628343471596977139160, 1.70839354067922653325319311489, 2.83305098841707394915484611228, 3.34141843922185728474017790444, 4.75835796946960270432925923708, 5.26725862145207982773369411809, 6.09660806336430263740514837573, 7.30269017111645724957058487494, 7.969344766486127905846603490223, 8.758758086969281188205838990491
