L(s) = 1 | − 2-s + (1.22 + 1.22i)3-s + 4-s + (−1.22 − 1.22i)6-s + (2.44 − i)7-s − 8-s + 2.99i·9-s + (1.22 + 1.22i)12-s + 2.44·13-s + (−2.44 + i)14-s + 16-s − 4.89i·17-s − 2.99i·18-s + 2.44i·19-s + (4.22 + 1.77i)21-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.707 + 0.707i)3-s + 0.5·4-s + (−0.499 − 0.499i)6-s + (0.925 − 0.377i)7-s − 0.353·8-s + 0.999i·9-s + (0.353 + 0.353i)12-s + 0.679·13-s + (−0.654 + 0.267i)14-s + 0.250·16-s − 1.18i·17-s − 0.707i·18-s + 0.561i·19-s + (0.921 + 0.387i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.724388002\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.724388002\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.44 + i)T \) |
good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4.89iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4.89T + 97T^{2} \) |
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
−9.962419397788195053873239959593, −8.970890152456223872422057987855, −8.647867849458358158612304596594, −7.61151363219966213292181741784, −7.08652509664568977888300976947, −5.59092940378462669457308489535, −4.73518783674921527057653967607, −3.66683570543446245681076535867, −2.59988064214740395969927494551, −1.29218802529303997066522246315,
1.12903573700320245060716479490, 2.10598673421650580656204028446, 3.19488000423820397659244435435, 4.47247933850251196683150321049, 5.85903615566645939506625499155, 6.59405574380998347546581101998, 7.65585433573267170984332915166, 8.158653865810366086921907921221, 8.881054909710249681043958768185, 9.489858376281511650677689616568