| L(s) = 1 | + 4i·5-s − 4i·7-s − 16·11-s − 2i·13-s − 24·17-s + 24·19-s + 32i·23-s + 9·25-s − 44i·29-s + 52i·31-s + 16·35-s − 18i·37-s + 8·41-s − 56·43-s − 32i·47-s + ⋯ |
| L(s) = 1 | + 0.800i·5-s − 0.571i·7-s − 1.45·11-s − 0.153i·13-s − 1.41·17-s + 1.26·19-s + 1.39i·23-s + 0.359·25-s − 1.51i·29-s + 1.67i·31-s + 0.457·35-s − 0.486i·37-s + 0.195·41-s − 1.30·43-s − 0.680i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.315923735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315923735\) |
| \(L(2)\) |
|
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 | \( 1 - 4iT - 25T^{2} \) |
| 7 | \( 1 + 4iT - 49T^{2} \) |
| 11 | \( 1 + 16T + 121T^{2} \) |
| 13 | \( 1 + 2iT - 169T^{2} \) |
| 17 | \( 1 + 24T + 289T^{2} \) |
| 19 | \( 1 - 24T + 361T^{2} \) |
| 23 | \( 1 - 32iT - 529T^{2} \) |
| 29 | \( 1 + 44iT - 841T^{2} \) |
| 31 | \( 1 - 52iT - 961T^{2} \) |
| 37 | \( 1 + 18iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 56T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 32T + 3.48e3T^{2} \) |
| 61 | \( 1 + 62iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 80T + 4.48e3T^{2} \) |
| 71 | \( 1 + 128iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 66T + 5.32e3T^{2} \) |
| 79 | \( 1 - 20iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 16T + 6.88e3T^{2} \) |
| 89 | \( 1 + 144T + 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−8.634139466132426938120027236310, −7.82370628990227040201864058061, −7.19990329581567557492871508024, −6.58643701833148806334060383167, −5.47624942912492165009026177904, −4.86353854026262915077933327359, −3.66105943761303117820425501390, −2.94078720553002753792694425607, −1.95730085980960235179380175873, −0.41041227621845883016139404210,
0.818109151189451698056032792845, 2.21425204502679104608040221773, 2.92315899379041569433953463895, 4.28898652742266672214473141129, 5.01876665500674753136083694001, 5.57310933594971628906782703117, 6.60939513076052826859698496447, 7.44193098919170421722713360005, 8.376135364907829871833171529786, 8.759511770890282911929815815448
