| L(s) = 1 | + i·5-s + 3.46i·11-s − 3.46i·13-s + 3.46·17-s − 4i·19-s − 25-s − 6i·29-s + 3.46·31-s + 3.46i·37-s + 6.92·41-s + 4i·43-s + 12·47-s − 7·49-s − 6i·53-s − 3.46·55-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.04i·11-s − 0.960i·13-s + 0.840·17-s − 0.917i·19-s − 0.200·25-s − 1.11i·29-s + 0.622·31-s + 0.569i·37-s + 1.08·41-s + 0.609i·43-s + 1.75·47-s − 49-s − 0.824i·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.853730215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853730215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 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
−8.800508824608342217714777630129, −7.84553102705627982139836333688, −7.44058230370264450913075236091, −6.54557554875673890112314262238, −5.77493397539520065847232989202, −4.90691471623013094618205249913, −4.09532360899141226365697657076, −3.02026343506896748650631050856, −2.29819271967139955006921192372, −0.869037671327452069142446685465,
0.842738030592268936645771297974, 1.92983916610644226779839795130, 3.19509498107782360727400721518, 3.93475630189780193865673738879, 4.88007761284519062587512421846, 5.74284816202495932761971317163, 6.31037189464703243422293775745, 7.35188580184209945372676028554, 8.010354514185014335600785097374, 8.829823358113761150033192325106
