L(s) = 1 | − 1.53i·2-s + i·3-s − 0.369·4-s + 1.53·6-s − 3.87i·7-s − 2.51i·8-s − 9-s + 1.24·11-s − 0.369i·12-s − i·13-s − 5.97·14-s − 4.60·16-s − 0.659i·17-s + 1.53i·18-s − 6.97·19-s + ⋯ |
L(s) = 1 | − 1.08i·2-s + 0.577i·3-s − 0.184·4-s + 0.628·6-s − 1.46i·7-s − 0.887i·8-s − 0.333·9-s + 0.376·11-s − 0.106i·12-s − 0.277i·13-s − 1.59·14-s − 1.15·16-s − 0.160i·17-s + 0.362i·18-s − 1.59·19-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
\(L(1)\) |
\(\approx\) |
\(0.325766 - 1.37996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.325766 - 1.37996i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.53iT - 2T^{2} \) |
| 7 | \( 1 + 3.87iT - 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 17 | \( 1 + 0.659iT - 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 1.55iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 2.29iT - 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 8.20iT - 43T^{2} \) |
| 47 | \( 1 + 1.53iT - 47T^{2} \) |
| 53 | \( 1 + 1.44iT - 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.92iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 6.20iT - 73T^{2} \) |
| 79 | \( 1 - 0.474T + 79T^{2} \) |
| 83 | \( 1 - 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 11.6iT - 97T^{2} \) |
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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
−10.15935129583940882396361596796, −9.117278863418707994112089885648, −8.136596755868652415993602835105, −7.01653609203166301330520580924, −6.36981558845086305472950148271, −4.82692553837020464877217494811, −4.00662912223271509227491943278, −3.36484439910107361669283994012, −2.03524288369887256779543882800, −0.62893996857189094187064835328,
1.91060319002650878558321971596, 2.79303442209984698353373126784, 4.53785260440403047729372683806, 5.49338314734323632599618403034, 6.45783410584167674913199245098, 6.57191462534130008080500839318, 7.930996306495751394050884997269, 8.505529903474084940452187192135, 9.039227668272106499854948977192, 10.29796123833495086515785606610