L(s) = 1 | + 0.692i·3-s + (2.22 + 0.245i)5-s + (0.343 + 0.343i)7-s + 2.52·9-s + (−0.843 − 0.843i)11-s − 3.68·13-s + (−0.169 + 1.53i)15-s + (0.412 + 0.412i)17-s + (5.37 + 5.37i)19-s + (−0.238 + 0.238i)21-s + (3.08 − 3.08i)23-s + (4.87 + 1.09i)25-s + 3.82i·27-s + (−4.22 + 4.22i)29-s − 8.75i·31-s + ⋯ |
L(s) = 1 | + 0.399i·3-s + (0.993 + 0.109i)5-s + (0.129 + 0.129i)7-s + 0.840·9-s + (−0.254 − 0.254i)11-s − 1.02·13-s + (−0.0438 + 0.397i)15-s + (0.0999 + 0.0999i)17-s + (1.23 + 1.23i)19-s + (−0.0519 + 0.0519i)21-s + (0.643 − 0.643i)23-s + (0.975 + 0.218i)25-s + 0.735i·27-s + (−0.785 + 0.785i)29-s − 1.57i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52717 + 0.403798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52717 + 0.403798i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.22 - 0.245i)T \) |
good | 3 | \( 1 - 0.692iT - 3T^{2} \) |
| 7 | \( 1 + (-0.343 - 0.343i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.843 + 0.843i)T + 11iT^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 + (-0.412 - 0.412i)T + 17iT^{2} \) |
| 19 | \( 1 + (-5.37 - 5.37i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.08 + 3.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.22 - 4.22i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.75iT - 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 2.54iT - 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + (4.56 - 4.56i)T - 47iT^{2} \) |
| 53 | \( 1 + 6.07iT - 53T^{2} \) |
| 59 | \( 1 + (-7.33 + 7.33i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.81 + 4.81i)T + 61iT^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 + (6.87 + 6.87i)T + 73iT^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 - 1.10T + 89T^{2} \) |
| 97 | \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \) |
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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
−11.69009659739120144230737646089, −10.49887015917397794715854988202, −9.893828113318169842023679081729, −9.209975275047172022090509321904, −7.85586538960897726344169556808, −6.87180494866536375195626418793, −5.63055586715815956037406147513, −4.81836443213003672788673296049, −3.30089196054522489401799715617, −1.78435324985501658386270478951,
1.46677832997553730242222070232, 2.84334167927277081386320165730, 4.69533126435310035574399400509, 5.51009096167842390183302879042, 6.98623654368914753880575925344, 7.38319109015223060651651898184, 8.923319806149586855662497345611, 9.740797076142829179457997798699, 10.43188378764993112506624417432, 11.68064131267422026115071911250