| L(s) = 1 | − i·2-s + 0.683i·3-s − 4-s + 0.683·6-s − 2.16i·7-s + i·8-s + 2.53·9-s − 4.32·11-s − 0.683i·12-s + 5.64i·13-s − 2.16·14-s + 16-s + 4.16i·17-s − 2.53i·18-s + 1.36·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.394i·3-s − 0.5·4-s + 0.279·6-s − 0.818i·7-s + 0.353i·8-s + 0.844·9-s − 1.30·11-s − 0.197i·12-s + 1.56i·13-s − 0.578·14-s + 0.250·16-s + 1.00i·17-s − 0.596i·18-s + 0.313·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.372748726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372748726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.683iT - 3T^{2} \) |
| 7 | \( 1 + 2.16iT - 7T^{2} \) |
| 11 | \( 1 + 4.32T + 11T^{2} \) |
| 13 | \( 1 - 5.64iT - 13T^{2} \) |
| 17 | \( 1 - 4.16iT - 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 0.683iT - 23T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 - 6.32iT - 37T^{2} \) |
| 41 | \( 1 + 2.32T + 41T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 8.49iT - 53T^{2} \) |
| 59 | \( 1 + 2.27T + 59T^{2} \) |
| 61 | \( 1 - 2.68T + 61T^{2} \) |
| 67 | \( 1 - 2.73iT - 67T^{2} \) |
| 71 | \( 1 - 2.73T + 71T^{2} \) |
| 73 | \( 1 - 15.2iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 6.63iT - 83T^{2} \) |
| 89 | \( 1 - 5.28T + 89T^{2} \) |
| 97 | \( 1 + 0.987iT - 97T^{2} \) |
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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
−9.845890839547831618087251555095, −8.971989988104932988934598859718, −8.051452570491842452156830438051, −7.24061090036187910840482882648, −6.35172587878355361781679679804, −5.05800157127252964085172912542, −4.35338849204517547796911813732, −3.69292942183941072289349829069, −2.40307484277992064745448874600, −1.25250922116294137712448783863,
0.61445261276570403090813048055, 2.35000423964756110147751867795, 3.29277572000471531368921264672, 4.80566074084754804069618905012, 5.33382069579535994248421875659, 6.16803025583159897921709787595, 7.16957112876037427720656837297, 7.85226308638343924574990786035, 8.318177716147300571646284224798, 9.466904462006806001892086017245
