L(s) = 1 | − i·2-s − 4-s − 2i·7-s + i·8-s + 11-s − 4i·13-s − 2·14-s + 16-s − 6i·17-s + 4·19-s − i·22-s − 6i·23-s − 4·26-s + 2i·28-s + 6·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s + 0.301·11-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.917·19-s − 0.213i·22-s − 1.25i·23-s − 0.784·26-s + 0.377i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.746339886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746339886\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.238414568331591139577640142010, −7.16686295938038070670790008547, −6.66337877143013355261865359864, −5.58318888448732536257409824816, −4.79320397724313851884403347112, −4.26833418064369118492580917684, −2.98717809726799514704443415120, −2.85540869092853404967424156546, −1.23962517798520136727570334496, −0.55489720668283507282439958680,
1.25051033714618332800399908691, 2.25654632263624138682425558260, 3.44821593090815095482383272144, 4.16205507094380973754079920528, 5.02942439977974422593969569530, 5.80604571111637262822386563681, 6.32538742300598588344515614945, 7.10209218398243938230348323668, 7.78924244967713017272742860667, 8.664782909518632297295912105324