L(s) = 1 | − 2i·2-s − 0.589i·3-s − 4·4-s − 1.17·6-s − 18.5i·7-s + 8i·8-s + 26.6·9-s + 47.9·11-s + 2.35i·12-s − 42.3i·13-s − 37.0·14-s + 16·16-s + 1.70i·17-s − 53.3i·18-s − 21.4·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.113i·3-s − 0.5·4-s − 0.0802·6-s − 0.999i·7-s + 0.353i·8-s + 0.987·9-s + 1.31·11-s + 0.0567i·12-s − 0.903i·13-s − 0.706·14-s + 0.250·16-s + 0.0243i·17-s − 0.697i·18-s − 0.258·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.453531310\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.453531310\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23iT \) |
good | 3 | \( 1 + 0.589iT - 27T^{2} \) |
| 7 | \( 1 + 18.5iT - 343T^{2} \) |
| 11 | \( 1 - 47.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 1.70iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 21.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 57.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 7.85iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 465.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 182. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 449. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 368. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 377.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 849.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 92.3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 439. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3iT - 9.12e5T^{2} \) |
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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
−9.433560101887238996405847723990, −8.394987768573654257755624199026, −7.49918632085029321058923975133, −6.77256846938113285112930540390, −5.74295023254039612086000205484, −4.32535530615805669323468892817, −4.05959195486291046681255154037, −2.81865737087088163279932728553, −1.40580023287752264160305148522, −0.74529133232682284213006923173,
1.11412862230001541476485473563, 2.32893257719290736609711920021, 3.87957334810724052990450028771, 4.48118742132648312696760975409, 5.56569325665969425353949590754, 6.52697866402448722593196527997, 6.93664370093997847839597434554, 8.113597192230950849152498172810, 8.902569645812813294655374238136, 9.469154946023111902872192512108