| L(s) = 1 | − 5·3-s + (5 + 24.4i)5-s + (−35 + 34.2i)7-s − 56·9-s − 89·11-s − 5·13-s + (−25 − 122. i)15-s − 485·17-s − 220. i·19-s + (175 − 171. i)21-s − 700. i·23-s + (−575 + 244. i)25-s + 685·27-s + 191·29-s + 1.05e3i·31-s + ⋯ |
| L(s) = 1 | − 0.555·3-s + (0.200 + 0.979i)5-s + (−0.714 + 0.699i)7-s − 0.691·9-s − 0.735·11-s − 0.0295·13-s + (−0.111 − 0.544i)15-s − 1.67·17-s − 0.610i·19-s + (0.396 − 0.388i)21-s − 1.32i·23-s + (−0.920 + 0.391i)25-s + 0.939·27-s + 0.227·29-s + 1.09i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4836749103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4836749103\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-5 - 24.4i)T \) |
| 7 | \( 1 + (35 - 34.2i)T \) |
| good | 3 | \( 1 + 5T + 81T^{2} \) |
| 11 | \( 1 + 89T + 1.46e4T^{2} \) |
| 13 | \( 1 + 5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 485T + 8.35e4T^{2} \) |
| 19 | \( 1 + 220. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 700. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 191T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.05e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.63e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.91e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 377. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.19e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.58e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.93e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.04e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.65e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.56e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.99e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 808. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.23e3T + 8.85e7T^{2} \) |
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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
−10.34762334649837947492760897215, −9.169638534159307728186736841545, −8.467877286345460523148982131305, −7.07525396134047596362530780915, −6.39870601373897775569000562004, −5.68137052731884766260524664342, −4.53470618314882513731078860753, −2.92075571077266051683308480666, −2.42249723715861735406714283384, −0.21156945016321496001411483416,
0.62467970414361595377804218048, 2.17088039335104699313787041114, 3.64771237766723814291111421518, 4.74548992969961015363109269051, 5.63540830055588176351237387591, 6.44907595489095960153264639961, 7.58366518001716211441694331732, 8.558946248295354205110895279237, 9.390254149815530130981866725708, 10.26736103750383570086234506611
