L(s) = 1 | − 2-s + (−0.707 − 0.707i)3-s + 4-s + (2.23 − 0.0221i)5-s + (0.707 + 0.707i)6-s + (−0.281 − 0.281i)7-s − 8-s + 1.00i·9-s + (−2.23 + 0.0221i)10-s + 1.64i·11-s + (−0.707 − 0.707i)12-s + 3.15·13-s + (0.281 + 0.281i)14-s + (−1.59 − 1.56i)15-s + 16-s + 3.44i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.999 − 0.00990i)5-s + (0.288 + 0.288i)6-s + (−0.106 − 0.106i)7-s − 0.353·8-s + 0.333i·9-s + (−0.707 + 0.00700i)10-s + 0.496i·11-s + (−0.204 − 0.204i)12-s + 0.875·13-s + (0.0752 + 0.0752i)14-s + (−0.412 − 0.404i)15-s + 0.250·16-s + 0.835i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.210409094\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.210409094\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-2.23 + 0.0221i)T \) |
| 37 | \( 1 + (-0.925 + 6.01i)T \) |
good | 7 | \( 1 + (0.281 + 0.281i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.64iT - 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 - 3.44iT - 17T^{2} \) |
| 19 | \( 1 + (1.85 - 1.85i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 + (-2.67 - 2.67i)T + 29iT^{2} \) |
| 31 | \( 1 + (6.77 - 6.77i)T - 31iT^{2} \) |
| 41 | \( 1 - 8.72iT - 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + (0.752 + 0.752i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.88 + 3.88i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.53 - 6.53i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.144 - 0.144i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.492 - 0.492i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + (-0.804 - 0.804i)T + 73iT^{2} \) |
| 79 | \( 1 + (-2.86 + 2.86i)T - 79iT^{2} \) |
| 83 | \( 1 + (-3.45 + 3.45i)T - 83iT^{2} \) |
| 89 | \( 1 + (-12.5 - 12.5i)T + 89iT^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−9.947872280262607292763614453787, −9.013994536928556591861075006968, −8.417990784278047273570661149597, −7.30317999804998258143115947119, −6.55786652449716042070282305626, −5.90805757984863132189347017170, −4.95197309075908995615778456038, −3.47772224080578416927232331452, −2.09589553875311115649236893942, −1.22806458449160464848578474904,
0.822403670398993610246555534925, 2.26620678925808932095748352347, 3.38446007639234779135628521154, 4.76978354533832239905396824721, 5.76296358294706418628172785321, 6.33817494168126018361569654094, 7.25467028924830245720259469829, 8.425559018114580560843197224291, 9.204594838817826378947484874135, 9.609945017177219240879962253479