L(s) = 1 | − 2.56i·3-s + 2.56i·7-s − 3.56·9-s − 11-s − 2i·13-s − 0.561i·17-s + 2.56·19-s + 6.56·21-s + 5.12i·23-s + 1.43i·27-s − 9.68·29-s − 6.56·31-s + 2.56i·33-s − 5.68i·37-s − 5.12·39-s + ⋯ |
L(s) = 1 | − 1.47i·3-s + 0.968i·7-s − 1.18·9-s − 0.301·11-s − 0.554i·13-s − 0.136i·17-s + 0.587·19-s + 1.43·21-s + 1.06i·23-s + 0.276i·27-s − 1.79·29-s − 1.17·31-s + 0.445i·33-s − 0.934i·37-s − 0.820·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7615732443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7615732443\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2.56iT - 3T^{2} \) |
| 7 | \( 1 - 2.56iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 0.561iT - 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 5.12iT - 23T^{2} \) |
| 29 | \( 1 + 9.68T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 5.68iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 13.1iT - 47T^{2} \) |
| 53 | \( 1 + 4.56iT - 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + 6.24iT - 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.12iT - 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 7.12iT - 97T^{2} \) |
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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
−8.241510184942919508912097308787, −7.56503001018456168706660553695, −7.33908637874224176198679453377, −6.21072393731730160942941452721, −5.72370614529849194671141234188, −5.12666041615254361737532666183, −3.72827192439604540352306841436, −2.79817130821994376818001268976, −2.05124672288470713072205294571, −1.17765031794342621784151060168,
0.21531089094521229089344144852, 1.80867841030892429950969858239, 3.07115207355862851677081857060, 3.91803491941726052986291776675, 4.24164060576108072551550875449, 5.20210533202993948726125663401, 5.73043128293770188095925728330, 6.96713589044045176645298248924, 7.37375530626784333813826250266, 8.481903778283866838241340102626