L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s + 4.56·7-s + 8-s − 9-s + 2.56i·11-s + i·12-s + (−3.56 + 0.561i)13-s + 4.56·14-s + 16-s + 5.68i·17-s − 18-s + 4.68i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.72·7-s + 0.353·8-s − 0.333·9-s + 0.772i·11-s + 0.288i·12-s + (−0.987 + 0.155i)13-s + 1.21·14-s + 0.250·16-s + 1.37i·17-s − 0.235·18-s + 1.07i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.134780398\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.134780398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.56 - 0.561i)T \) |
good | 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 - 2.56iT - 11T^{2} \) |
| 17 | \( 1 - 5.68iT - 17T^{2} \) |
| 19 | \( 1 - 4.68iT - 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 1.43iT - 31T^{2} \) |
| 37 | \( 1 - 0.438T + 37T^{2} \) |
| 41 | \( 1 - 0.438iT - 41T^{2} \) |
| 43 | \( 1 - 2.87iT - 43T^{2} \) |
| 47 | \( 1 + 0.123T + 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 + 5.43iT - 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 3.12iT - 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−9.448298775541371273926893435599, −8.247677853120741730432083161499, −7.976203878126542757912087228058, −6.93866683568875392276277256900, −5.92739318247823280687298326774, −5.06829470931469848649815236220, −4.50514807680803187153381606090, −3.86354345729159526014236897649, −2.41509112563544248808065595068, −1.63433480163886803347015552337,
0.924285400919668761143694720428, 2.15164465023259595211167077661, 2.94617678224004847199214692661, 4.25449104993498039730157823362, 5.24407233698133812719026404565, 5.38522889979832200756657608483, 6.83775924364765098792481916147, 7.34523999717992977688511822178, 8.076563003638453624339274561264, 8.825013793256585768837836394750