| L(s) = 1 | + (−1.22 + 1.22i)3-s + (2.44 + 2.44i)7-s − 2.99i·9-s + (4.89 − 4.89i)13-s − 8i·19-s − 5.99·21-s + (3.67 + 3.67i)27-s − 4·31-s + (4.89 + 4.89i)37-s + 11.9i·39-s + (7.34 − 7.34i)43-s + 4.99i·49-s + (9.79 + 9.79i)57-s + 14·61-s + (7.34 − 7.34i)63-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.925 + 0.925i)7-s − 0.999i·9-s + (1.35 − 1.35i)13-s − 1.83i·19-s − 1.30·21-s + (0.707 + 0.707i)27-s − 0.718·31-s + (0.805 + 0.805i)37-s + 1.92i·39-s + (1.12 − 1.12i)43-s + 0.714i·49-s + (1.29 + 1.29i)57-s + 1.79·61-s + (0.925 − 0.925i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492924303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492924303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-4.89 + 4.89i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-7.34 + 7.34i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + (-2.44 - 2.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (9.79 - 9.79i)T - 73iT^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-9.79 - 9.79i)T + 97iT^{2} \) |
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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
−9.835223324398076957351019596665, −8.837152012640085170406904227496, −8.469349080978693308946044805363, −7.26804563348882440188022671207, −6.14637066227117834742440675677, −5.47592838312825359066453933285, −4.83530600642989268922747141246, −3.72538329168478721571651129590, −2.58936030361621522850748381655, −0.894976030896496959444208243933,
1.16354033796269817599740498782, 1.91299801198480199484279969546, 3.81545736973331554632601859933, 4.48676846074074524970299303122, 5.69729957882976061251911394362, 6.33360383097670384700274976645, 7.30012201275111563307110199381, 7.907379914827237304591740676944, 8.707942425074359205205208186687, 9.862919084795743287044580317453
