L(s) = 1 | + (−1.75 + 0.950i)2-s + (2.19 − 3.34i)4-s − 3.98·7-s + (−0.677 + 7.97i)8-s + 8.39i·11-s + 9.77i·13-s + (7.02 − 3.79i)14-s + (−6.38 − 14.6i)16-s + 12.1i·17-s − 17.3i·19-s + (−7.97 − 14.7i)22-s − 2.97·23-s + (−9.28 − 17.1i)26-s + (−8.74 + 13.3i)28-s + 51.6·29-s + ⋯ |
L(s) = 1 | + (−0.879 + 0.475i)2-s + (0.548 − 0.836i)4-s − 0.569·7-s + (−0.0847 + 0.996i)8-s + 0.763i·11-s + 0.751i·13-s + (0.501 − 0.270i)14-s + (−0.399 − 0.916i)16-s + 0.714i·17-s − 0.914i·19-s + (−0.362 − 0.671i)22-s − 0.129·23-s + (−0.357 − 0.661i)26-s + (−0.312 + 0.476i)28-s + 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08649682094\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08649682094\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.75 - 0.950i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.98T + 49T^{2} \) |
| 11 | \( 1 - 8.39iT - 121T^{2} \) |
| 13 | \( 1 - 9.77iT - 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 + 17.3iT - 361T^{2} \) |
| 23 | \( 1 + 2.97T + 529T^{2} \) |
| 29 | \( 1 - 51.6T + 841T^{2} \) |
| 31 | \( 1 + 30.7iT - 961T^{2} \) |
| 37 | \( 1 + 19.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 21.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 94.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 132. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 48.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 140.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 18.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 15iT - 9.40e3T^{2} \) |
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
−10.02325975164776047225544706489, −9.671012737687540626062283236775, −8.687512307614779721560834918756, −8.001289899351836057444409829714, −6.76979502561475803508953659119, −6.60740099172675824983959615758, −5.31501403886043556694387779405, −4.29448683491163216286742235818, −2.74816803755217287504068705802, −1.57007016185866881897297823843,
0.03909852581593062371443644906, 1.34430826028540069093121765845, 2.90568749729262481227140526457, 3.45503660566136909831454792964, 4.95190486971611572716195228503, 6.23331931630231051513190801995, 6.90878675709365709358549848577, 8.114821736251537098140641160617, 8.472499130336864927161103180559, 9.599637299989303438808500028826