L(s) = 1 | + (0.309 − 0.951i)4-s + (1.34 + 0.437i)7-s + (0.831 + 1.14i)13-s + (−0.809 − 0.587i)16-s + (−1.34 + 0.437i)19-s + (−0.309 − 0.951i)25-s + (0.831 − 1.14i)28-s − 1.41i·43-s + (0.809 + 0.587i)49-s + (1.34 − 0.437i)52-s + (−0.831 + 1.14i)61-s + (−0.809 + 0.587i)64-s + (−1.34 − 0.437i)73-s + 1.41i·76-s + (−0.831 − 1.14i)79-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)4-s + (1.34 + 0.437i)7-s + (0.831 + 1.14i)13-s + (−0.809 − 0.587i)16-s + (−1.34 + 0.437i)19-s + (−0.309 − 0.951i)25-s + (0.831 − 1.14i)28-s − 1.41i·43-s + (0.809 + 0.587i)49-s + (1.34 − 0.437i)52-s + (−0.831 + 1.14i)61-s + (−0.809 + 0.587i)64-s + (−1.34 − 0.437i)73-s + 1.41i·76-s + (−0.831 − 1.14i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.255164074\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255164074\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
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
−10.26049455229931172736442924705, −8.992513199378266023443944369791, −8.597248280288605353954266199816, −7.52703240013495333953531927630, −6.45530849275597084072011207611, −5.86529299030125554315179142403, −4.82585345691841437999834019677, −4.09836508095924963361008301762, −2.26437847382682777689274174188, −1.56006069287549804082634985708,
1.61142701254934689006492194806, 2.90227370705713524909050056627, 3.95937128375413067679711817758, 4.77646640401081482785739911088, 5.91473680511968734111925871522, 6.94690017419879446322267815243, 7.86867275712211780736987152490, 8.223221898001873622749823420390, 9.047133809835957975543022214989, 10.35922380486440221924224025000