| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 2.61i·7-s + i·8-s − 9-s + 3.61·11-s + i·12-s − 6.47i·13-s − 2.61·14-s + 16-s − 1.23i·17-s + i·18-s + 5.70·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.989i·7-s + 0.353i·8-s − 0.333·9-s + 1.09·11-s + 0.288i·12-s − 1.79i·13-s − 0.699·14-s + 0.250·16-s − 0.299i·17-s + 0.235i·18-s + 1.30·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.791519785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.791519785\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.61iT - 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 + 6.47iT - 13T^{2} \) |
| 17 | \( 1 + 1.23iT - 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + 7.70iT - 43T^{2} \) |
| 47 | \( 1 + 1.70iT - 47T^{2} \) |
| 53 | \( 1 + 2.09iT - 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 1.52iT - 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.52iT - 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 - 2.14iT - 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 - 3.38iT - 97T^{2} \) |
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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
−7.960228139042282295085358663596, −7.54124481345270894115278222579, −6.85019651332769386581854847655, −5.70357608539399250199509653455, −5.25022517195596392602221165246, −3.91073738601513652425396170018, −3.51872437844917882741318008169, −2.48181653355048834961349957212, −1.23971743193916980544054294320, −0.61612158852901280846381102638,
1.37937092007974387497095449482, 2.58567629558690599962904283517, 3.71141532505627431820454143280, 4.41420496348118031964935637685, 5.10394689678368055648557169948, 6.08978075252129906517765558394, 6.46440595074770349108878293805, 7.33754835059570422794632754655, 8.286531454694500955936394233503, 8.946775747777652159682209443908
