L(s) = 1 | + 3-s + 7-s + 13-s − 1.41i·17-s + 1.41i·19-s + 21-s + 1.41i·23-s − 25-s − 27-s − 1.41i·31-s − 1.41i·37-s + 39-s − 41-s + 43-s + 1.41i·47-s + ⋯ |
L(s) = 1 | + 3-s + 7-s + 13-s − 1.41i·17-s + 1.41i·19-s + 21-s + 1.41i·23-s − 25-s − 27-s − 1.41i·31-s − 1.41i·37-s + 39-s − 41-s + 43-s + 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.780551278\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780551278\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - 1.41iT - T^{2} \) |
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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
−9.312736393864224081467563572169, −8.405172274389169213269541403539, −7.82840153875612647320960060190, −7.37468411533620249800865858194, −5.94267811948929661101334852475, −5.44065126750227160110801092205, −4.15949182907551378166290375170, −3.54494928420300010174921765875, −2.42780047548210021659517563906, −1.50502347294976308315535140653,
1.50031052699412350748376119261, 2.44972261937815343592894593931, 3.46469875050591817784900723572, 4.29525278779773898392902697755, 5.21485591405639881967649004281, 6.22539594181083112204551082304, 7.04022234099292925304177838587, 8.155892555985593367836941257507, 8.464597306877219957717274200353, 8.906660853001429182504110136387