| L(s) = 1 | − i·2-s + i·3-s + 4-s + 6-s + 3i·7-s − 3i·8-s − 9-s − 11-s + i·12-s + i·13-s + 3·14-s − 16-s + 5i·17-s + i·18-s + 8·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + 0.408·6-s + 1.13i·7-s − 1.06i·8-s − 0.333·9-s − 0.301·11-s + 0.288i·12-s + 0.277i·13-s + 0.801·14-s − 0.250·16-s + 1.21i·17-s + 0.235i·18-s + 1.83·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79410 + 0.423531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79410 + 0.423531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 11iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.953219325022572253019619379447, −9.610572909105518852360956890047, −8.548589806551599059976766094615, −7.69713105689630168693199699420, −6.48066467962313670565829759637, −5.74135727289181725149398634475, −4.72309955422586542185131622997, −3.44831572882204286425184494551, −2.74840290685116707800225510351, −1.53357364983604927373442074658,
0.902528051191669292261780888325, 2.42895203304094878911833113373, 3.52997842255030820555157743329, 5.02701032876611107425291241719, 5.70509225015289243159030107494, 6.88062157319066545505867244925, 7.34426558003431121544280184614, 7.81579865387732758228568223813, 8.931870849238938903462290077281, 9.993365486017427421181187407919
