| L(s) = 1 | − 2.67i·2-s + 2.48i·3-s − 5.15·4-s + 6.63·6-s − i·7-s + 8.44i·8-s − 3.15·9-s − 11-s − 12.7i·12-s − 5.83i·13-s − 2.67·14-s + 12.2·16-s + 5.44i·17-s + 8.44i·18-s + 1.35·19-s + ⋯ |
| L(s) = 1 | − 1.89i·2-s + 1.43i·3-s − 2.57·4-s + 2.70·6-s − 0.377i·7-s + 2.98i·8-s − 1.05·9-s − 0.301·11-s − 3.69i·12-s − 1.61i·13-s − 0.714·14-s + 3.06·16-s + 1.32i·17-s + 1.99i·18-s + 0.309·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4896096374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4896096374\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67iT - 2T^{2} \) |
| 3 | \( 1 - 2.48iT - 3T^{2} \) |
| 13 | \( 1 + 5.83iT - 13T^{2} \) |
| 17 | \( 1 - 5.44iT - 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 3.19iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + 8.54iT - 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 + 5.89iT - 43T^{2} \) |
| 47 | \( 1 + 11.8iT - 47T^{2} \) |
| 53 | \( 1 - 0.231iT - 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 10.6iT - 83T^{2} \) |
| 89 | \( 1 + 7.22T + 89T^{2} \) |
| 97 | \( 1 + 0.836iT - 97T^{2} \) |
| 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
−9.085035961773667886130766411534, −8.503770515023587833848338029316, −7.58352474041309650630230809093, −5.68555323002137305527095657248, −5.17962366771172476260941093779, −4.22933656000982443989433708666, −3.56992958858925025792967828521, −3.04002041920816216941066277664, −1.70013175134360834801826608208, −0.19159736857780441853093724328,
1.34050116681965320086661755881, 2.82957248389742800511700881158, 4.43394815956325334041567838396, 5.04934960185115581240248295134, 6.19876807057220072902794433318, 6.49425833945783880489903179188, 7.29990285606081511635613411928, 7.69409334836481276788833281283, 8.605967731006244868785133559724, 9.133984533960537712305883850056
