L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 4.47i·7-s − i·8-s − 9-s − 5.23·11-s + i·12-s − 4.47i·13-s + 4.47·14-s + 16-s − 4i·17-s − i·18-s − 5.70·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.69i·7-s − 0.353i·8-s − 0.333·9-s − 1.57·11-s + 0.288i·12-s − 1.24i·13-s + 1.19·14-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s − 1.30·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2941499650\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2941499650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4.76iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 5.23iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 - 9.70iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47iT - 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 13.2iT - 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 0.472iT - 97T^{2} \) |
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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
−8.176451089274897619828960459311, −7.36299322738010090112652023789, −6.88362085959526673045347207237, −6.06641024144823618188781061942, −5.08426835419429211951036876639, −4.57847789755692117032663341918, −3.41477000264782655334941756020, −2.56586013284678452057470985085, −0.973172518120132657912481909096, −0.10061890932193294289648387655,
2.15476463719596557192755042117, 2.27870806276622124985344775793, 3.49376880310748834067853156357, 4.36458860521006432047482421844, 5.21145557924166855748307339910, 5.74350843135809622355478106236, 6.60205205974646345550272858913, 7.87287086399529270886678493574, 8.611440872025905094443441730240, 8.948814545491549441508535674218