| L(s) = 1 | + (2 − i)5-s + 3i·9-s + (−5 − 5i)13-s + (−5 + 5i)17-s + (3 − 4i)25-s + 4i·29-s + (5 − 5i)37-s + 8·41-s + (3 + 6i)45-s − 7i·49-s + (5 + 5i)53-s − 12·61-s + (−15 − 5i)65-s + (5 + 5i)73-s − 9·81-s + ⋯ |
| L(s) = 1 | + (0.894 − 0.447i)5-s + i·9-s + (−1.38 − 1.38i)13-s + (−1.21 + 1.21i)17-s + (0.600 − 0.800i)25-s + 0.742i·29-s + (0.821 − 0.821i)37-s + 1.24·41-s + (0.447 + 0.894i)45-s − i·49-s + (0.686 + 0.686i)53-s − 1.53·61-s + (−1.86 − 0.620i)65-s + (0.585 + 0.585i)73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01291 - 0.0455749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01291 - 0.0455749i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 3 | \( 1 - 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (5 + 5i)T + 13iT^{2} \) |
| 17 | \( 1 + (5 - 5i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-5 + 5i)T - 97iT^{2} \) |
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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
−14.32719116835131344129120718687, −13.13304650984422156221169451965, −12.60039612390521967293181515483, −10.85059261549145160635221462498, −10.06377174762825158964122584652, −8.755222109170443434349305797563, −7.53894588705916063426304420155, −5.87142516828485734671431238577, −4.75363631493516026702961808663, −2.33847888997287159193861359754,
2.49781743409028979957278186652, 4.59227006210364649411590411646, 6.29307059904154622423476335696, 7.18411973635644872290120913485, 9.215903905955930506680129574560, 9.674845215232649046593868163176, 11.22182095132037467626831895479, 12.18903620283370482013499177600, 13.52087354184667345390636775401, 14.36196059830176043760838360071
