| L(s) = 1 | + (−0.581 + 2.94i)3-s + 2.23i·5-s − 11.4·7-s + (−8.32 − 3.42i)9-s − 8.48i·11-s + 10·13-s + (−6.58 − 1.29i)15-s + 3.55i·17-s − 10.9·19-s + (6.67 − 33.8i)21-s − 17.6i·23-s − 5.00·25-s + (14.9 − 22.5i)27-s + 26.8i·29-s − 8·31-s + ⋯ |
| L(s) = 1 | + (−0.193 + 0.981i)3-s + 0.447i·5-s − 1.64·7-s + (−0.924 − 0.380i)9-s − 0.771i·11-s + 0.769·13-s + (−0.438 − 0.0866i)15-s + 0.209i·17-s − 0.577·19-s + (0.317 − 1.60i)21-s − 0.767i·23-s − 0.200·25-s + (0.552 − 0.833i)27-s + 0.925i·29-s − 0.258·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9527560442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9527560442\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.581 - 2.94i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 11.4T + 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 - 3.55iT - 289T^{2} \) |
| 19 | \( 1 + 10.9T + 361T^{2} \) |
| 23 | \( 1 + 17.6iT - 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 + 8T + 961T^{2} \) |
| 37 | \( 1 - 59.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 20.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 88.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 3.55iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 77.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 69.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 9.02T + 6.24e3T^{2} \) |
| 83 | \( 1 - 0.688iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.10iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 111.T + 9.40e3T^{2} \) |
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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
−9.913458455578473505249054426452, −9.037331577071289735831354483103, −8.484505094312299953972489752091, −7.03401539057959107411182036438, −6.17418003666519197271765918865, −5.72411601993127334564291905652, −4.23516664499465729506060221591, −3.48026193960884924941621258264, −2.71887408378686932376456410469, −0.41303600885086426766623521300,
0.877239607048254601442338705520, 2.26666734807404621267417377790, 3.36162972751559906463752608244, 4.56325354291431779951835249928, 6.00548593016296139782025689758, 6.24540600068557187937570186181, 7.31464288709961443292664986663, 8.016544999925871050344148270563, 9.182026354651063720680627309121, 9.614058177323165921888946921735
