L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 2i·7-s + i·8-s − 9-s + 4·11-s + i·12-s − i·13-s − 2·14-s + 16-s − 8i·17-s + i·18-s + 6·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.277i·13-s − 0.534·14-s + 0.250·16-s − 1.94i·17-s + 0.235i·18-s + 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.670077939\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.670077939\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−9.201104856662972777511994389222, −7.978446355489964497448618663604, −7.30537283870904236891044208165, −6.67978329200774810527219598183, −5.50212621464203298033577285997, −4.72266480019808520811833735313, −3.60063119545053609521505864535, −2.93028893992715558285640521905, −1.53772617662272086988607589872, −0.68983782162457881447637227363,
1.40899513089758659861335663546, 2.91304486896806243263708883554, 3.98457387246634535008300508367, 4.60241149650481552731327675004, 5.76272696304978056234992772970, 6.16734004398887691673910083590, 7.06738334483959098178049173674, 8.129834527985908072697693270206, 8.773660749318981909745276051264, 9.266852972621859906261207318532