L(s) = 1 | + (0.730 − 0.730i)3-s + (−2.41 − 1.08i)7-s + 1.93i·9-s + 5.95·11-s + (0.921 − 0.921i)13-s + (−2.02 − 2.02i)17-s + 3.29·19-s + (−2.55 + 0.970i)21-s + (−0.0544 − 0.0544i)23-s + (3.60 + 3.60i)27-s − 3.78i·29-s + 4.88i·31-s + (4.35 − 4.35i)33-s + (3.20 − 3.20i)37-s − 1.34i·39-s + ⋯ |
L(s) = 1 | + (0.421 − 0.421i)3-s + (−0.912 − 0.409i)7-s + 0.644i·9-s + 1.79·11-s + (0.255 − 0.255i)13-s + (−0.491 − 0.491i)17-s + 0.755·19-s + (−0.557 + 0.211i)21-s + (−0.0113 − 0.0113i)23-s + (0.693 + 0.693i)27-s − 0.703i·29-s + 0.876i·31-s + (0.757 − 0.757i)33-s + (0.527 − 0.527i)37-s − 0.215i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.956168260\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.956168260\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.41 + 1.08i)T \) |
good | 3 | \( 1 + (-0.730 + 0.730i)T - 3iT^{2} \) |
| 11 | \( 1 - 5.95T + 11T^{2} \) |
| 13 | \( 1 + (-0.921 + 0.921i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.02 + 2.02i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 + (0.0544 + 0.0544i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.78iT - 29T^{2} \) |
| 31 | \( 1 - 4.88iT - 31T^{2} \) |
| 37 | \( 1 + (-3.20 + 3.20i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (-1.60 - 1.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.64 + 2.64i)T + 47iT^{2} \) |
| 53 | \( 1 + (-9.16 - 9.16i)T + 53iT^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 + (-6.43 + 6.43i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 + 8.66i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.76iT - 79T^{2} \) |
| 83 | \( 1 + (-2.36 + 2.36i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 97 | \( 1 + (-2.05 - 2.05i)T + 97iT^{2} \) |
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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
−9.296568756393207773650479864263, −8.844837558041501603248230682951, −7.76245690019604182897310674752, −7.00748980355907762720662871600, −6.44836746128814599220679314465, −5.37233670199617090012330043187, −4.15409683891347768187413901106, −3.39875704078595773804869081104, −2.24894348441481092678335455417, −0.947279912695812496546305480354,
1.17321116014902063944422924275, 2.71392268404374791878176453128, 3.71645707885940442895735577237, 4.18385670615665042347978410415, 5.64003462200874379227371860383, 6.51466443509827728575648058027, 6.90750358524456075122329302715, 8.374353327035540902195431013788, 8.957349135395921951340642697475, 9.605003594248254809236728978433