L(s) = 1 | − 7.34i·5-s + 10.3·7-s + 8.48i·11-s + 10.3·13-s − 21.2i·17-s + 20·19-s + 14.6i·23-s − 29·25-s − 36.7i·29-s + 51.9·31-s − 76.3i·35-s + 41.5·37-s + 72.1i·41-s − 40·43-s + 73.4i·47-s + ⋯ |
L(s) = 1 | − 1.46i·5-s + 1.48·7-s + 0.771i·11-s + 0.799·13-s − 1.24i·17-s + 1.05·19-s + 0.638i·23-s − 1.15·25-s − 1.26i·29-s + 1.67·31-s − 2.18i·35-s + 1.12·37-s + 1.75i·41-s − 0.930·43-s + 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.938138782\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.938138782\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 7.34iT - 25T^{2} \) |
| 7 | \( 1 - 10.3T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 + 21.2iT - 289T^{2} \) |
| 19 | \( 1 - 20T + 361T^{2} \) |
| 23 | \( 1 - 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 36.7iT - 841T^{2} \) |
| 31 | \( 1 - 51.9T + 961T^{2} \) |
| 37 | \( 1 - 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 72.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40T + 1.84e3T^{2} \) |
| 47 | \( 1 - 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 100T + 4.48e3T^{2} \) |
| 71 | \( 1 - 73.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20T + 5.32e3T^{2} \) |
| 79 | \( 1 + 51.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 40T + 9.40e3T^{2} \) |
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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
−8.550804542988789536103271369893, −8.024562204353427978622663040821, −7.49283148184196244756930165469, −6.24425982317031920733190952157, −5.25442111131909582191539471001, −4.75703264555239349952117794339, −4.21735113280048398557077017268, −2.71983436806357122619580117049, −1.45118393424470884953513040512, −0.925977821629592830015472826714,
1.07958859717011251383113988173, 2.15434033059981774096778030668, 3.19768900578359794708159765777, 3.92003821644616479590952512481, 5.04537553364649923159894401258, 5.90115153763228743210634781437, 6.62419503527586086823507511556, 7.39952164232008791320150566427, 8.313788879267124889282492334839, 8.563278775736750276958398441946