L(s) = 1 | + (−0.846 + 1.13i)2-s + i·3-s + (−0.567 − 1.91i)4-s + (−2.22 − 0.235i)5-s + (−1.13 − 0.846i)6-s + (2.54 − 0.708i)7-s + (2.65 + 0.980i)8-s − 9-s + (2.14 − 2.31i)10-s − 5.27i·11-s + (1.91 − 0.567i)12-s + 2.60·13-s + (−1.35 + 3.48i)14-s + (0.235 − 2.22i)15-s + (−3.35 + 2.17i)16-s − 3.66·17-s + ⋯ |
L(s) = 1 | + (−0.598 + 0.801i)2-s + 0.577i·3-s + (−0.283 − 0.958i)4-s + (−0.994 − 0.105i)5-s + (−0.462 − 0.345i)6-s + (0.963 − 0.267i)7-s + (0.938 + 0.346i)8-s − 0.333·9-s + (0.679 − 0.733i)10-s − 1.59i·11-s + (0.553 − 0.163i)12-s + 0.722·13-s + (−0.361 + 0.932i)14-s + (0.0608 − 0.574i)15-s + (−0.839 + 0.544i)16-s − 0.889·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.900818 + 0.207700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.900818 + 0.207700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.846 - 1.13i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.22 + 0.235i)T \) |
| 7 | \( 1 + (-2.54 + 0.708i)T \) |
good | 11 | \( 1 + 5.27iT - 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 + 3.66T + 17T^{2} \) |
| 19 | \( 1 - 6.51T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 6.64T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 + 0.593iT - 37T^{2} \) |
| 41 | \( 1 - 8.01iT - 41T^{2} \) |
| 43 | \( 1 + 0.678T + 43T^{2} \) |
| 47 | \( 1 + 5.79iT - 47T^{2} \) |
| 53 | \( 1 + 1.21iT - 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 - 0.618iT - 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 + 6.11iT - 79T^{2} \) |
| 83 | \( 1 + 12.0iT - 83T^{2} \) |
| 89 | \( 1 - 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 0.241T + 97T^{2} \) |
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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
−11.23255142352500155804269152977, −10.44107166160817400353539838519, −9.073549065510602558772985337963, −8.481907929113574787616837554447, −7.82288922321342604762718462471, −6.70987326950155794733217096032, −5.47708770749356608591289397922, −4.62313123989436558969607996472, −3.41410874771009585851853440423, −0.905559347861139195978786763507,
1.30481160219601540550198542894, 2.63278913194652399237546856593, 4.06055196595382543507247134596, 5.02299868302937154117788401731, 7.01035869391602954705495407609, 7.50966687573431767772196421976, 8.444075190373201328361314375958, 9.181547017890901743898968457565, 10.45316911705079494799049633614, 11.27142160811017580408571537070