L(s) = 1 | − i·3-s − 2.08i·5-s + 5.03·7-s − 9-s − 0.828i·11-s − 2.94i·13-s − 2.08·15-s + 4.82·17-s − 2.82i·19-s − 5.03i·21-s − 4.16·23-s + 0.656·25-s + i·27-s + 7.97i·29-s + 5.03·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.932i·5-s + 1.90·7-s − 0.333·9-s − 0.249i·11-s − 0.817i·13-s − 0.538·15-s + 1.17·17-s − 0.648i·19-s − 1.09i·21-s − 0.869·23-s + 0.131·25-s + 0.192i·27-s + 1.48i·29-s + 0.903·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.134853902\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.134853902\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 + 2.08iT - 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 13 | \( 1 + 2.94iT - 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 7.97iT - 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 - 7.11iT - 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 + 12.4iT - 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 + 7.11iT - 61T^{2} \) |
| 67 | \( 1 - 2.34iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 + 3.17iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−8.829896554929302317549158590533, −8.439666077866384147343869016950, −7.82790228399456336309583217392, −7.05951283475142883992298980779, −5.68639400818008546376010092392, −5.18461560759201100762880480109, −4.44253341833751687339389847024, −3.05686248787806545164710262175, −1.68981595512792203993125454470, −0.954658607840750269847251182886,
1.55351942445669601712156309074, 2.57585877386610981717082450740, 3.86868175713437682495495675214, 4.55207881987771801260478455367, 5.46747147918781107231141743174, 6.33474183067394971495274164926, 7.47953847225646914434980891651, 7.987468803268846674086573011904, 8.759015761809235293312878814578, 9.978357738010674751680305473256