L(s) = 1 | + i·2-s − 4-s + (2 + i)5-s + i·7-s − i·8-s + (−1 + 2i)10-s + 4·11-s − 6i·13-s − 14-s + 16-s + 4i·17-s + (−2 − i)20-s + 4i·22-s − 4i·23-s + (3 + 4i)25-s + 6·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.377i·7-s − 0.353i·8-s + (−0.316 + 0.632i)10-s + 1.20·11-s − 1.66i·13-s − 0.267·14-s + 0.250·16-s + 0.970i·17-s + (−0.447 − 0.223i)20-s + 0.852i·22-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 1.17·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.164632005\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.164632005\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 13iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.261336616982137407345123798650, −8.502751786543832416514772547384, −7.86707618453461604827193536838, −6.72484996756684639802405871948, −6.22920376908770194754089773390, −5.60527531678230481104512019967, −4.63540676736281803147225319623, −3.50214956858527329589851689372, −2.51357256575273218945986234364, −1.09876514845993310767476532750,
1.06525973507405202567449390115, 1.87860502898605036926390410281, 3.01649718918609915686021115034, 4.28410542089247388937334648039, 4.65851816503273428830716536449, 5.91436659747403850494138391492, 6.60475945073188112973258094295, 7.48199602280395684057091193842, 8.720952197993778780745208800109, 9.308841049367887710786243275055