L(s) = 1 | + 2.82i·3-s − 5.00·9-s − 2.82i·11-s − 6·17-s + 8.48i·19-s − 5.65i·27-s + 8.00·33-s − 6·41-s − 8.48i·43-s − 7·49-s − 16.9i·51-s − 24·57-s − 14.1i·59-s − 8.48i·67-s + 2·73-s + ⋯ |
L(s) = 1 | + 1.63i·3-s − 1.66·9-s − 0.852i·11-s − 1.45·17-s + 1.94i·19-s − 1.08i·27-s + 1.39·33-s − 0.937·41-s − 1.29i·43-s − 49-s − 2.37i·51-s − 3.17·57-s − 1.84i·59-s − 1.03i·67-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 8.48iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.502557594773451952588524516234, −8.153479882046920004758441230925, −6.83715676619654464003352701775, −5.99732741971101711679774444358, −5.32518714829605328464478529861, −4.52204089047851344179858244256, −3.76099751794050762956616122236, −3.21449010169769752933809160542, −1.91035440040652120604335624898, 0,
1.26515106829252129867721578856, 2.24930582139775060100024243359, 2.83704722547804614558084846413, 4.33328258043778274527528992837, 5.04943558151247216641326473997, 6.16496569397807835252156188438, 6.83361235787672409599979364707, 7.13448648706446172564155887342, 7.973906474652379454068834179448