| L(s) = 1 | + (−1.58 + 4.74i)5-s + 3i·7-s + 9.48i·11-s + 21i·13-s − 12.6·17-s + 31·19-s + 22.1·23-s + (−20 − 15.0i)25-s + 47.4i·29-s + 16·31-s + (−14.2 − 4.74i)35-s + 27i·37-s − 47.4i·41-s + 48i·43-s + 12.6·47-s + ⋯ |
| L(s) = 1 | + (−0.316 + 0.948i)5-s + 0.428i·7-s + 0.862i·11-s + 1.61i·13-s − 0.744·17-s + 1.63·19-s + 0.962·23-s + (−0.800 − 0.600i)25-s + 1.63i·29-s + 0.516·31-s + (−0.406 − 0.135i)35-s + 0.729i·37-s − 1.15i·41-s + 1.11i·43-s + 0.269·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.575804409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575804409\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.58 - 4.74i)T \) |
| good | 7 | \( 1 - 3iT - 49T^{2} \) |
| 11 | \( 1 - 9.48iT - 121T^{2} \) |
| 13 | \( 1 - 21iT - 169T^{2} \) |
| 17 | \( 1 + 12.6T + 289T^{2} \) |
| 19 | \( 1 - 31T + 361T^{2} \) |
| 23 | \( 1 - 22.1T + 529T^{2} \) |
| 29 | \( 1 - 47.4iT - 841T^{2} \) |
| 31 | \( 1 - 16T + 961T^{2} \) |
| 37 | \( 1 - 27iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 47.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 12.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 41.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + T + 3.72e3T^{2} \) |
| 67 | \( 1 + 21iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 28.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 27iT - 5.32e3T^{2} \) |
| 79 | \( 1 - T + 6.24e3T^{2} \) |
| 83 | \( 1 + 110.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 113. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 93iT - 9.40e3T^{2} \) |
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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
−9.299517670251271803437503627346, −8.567824488023899396229593337206, −7.38554573052155404741237600961, −7.01900723288827215981748115446, −6.33786976491996886054093974411, −5.16241608849150930528975852097, −4.42180453147960649836156957675, −3.38837375895475496290333927180, −2.52404370638662269982798470188, −1.48630706301953191401225376768,
0.45428692620403345115813374308, 1.07858715357387718607577657506, 2.72853579326341594798068331471, 3.58020344766317732482899897263, 4.53155380859506497815264655145, 5.40048490831536428858916961246, 5.94708946174985058045013498214, 7.21061448928144323311936943483, 7.85385253735654769566826614630, 8.471604528785469544818426363408
