| L(s) = 1 | − 5.19·7-s + 23i·13-s − 36.3i·19-s + 19.0i·31-s + 26i·37-s − 22.5·43-s − 22·49-s − 121·61-s + 133.·67-s − 46i·73-s + 69.2i·79-s − 119. i·91-s − 167i·97-s + 69.2·103-s + 71·109-s + ⋯ |
| L(s) = 1 | − 0.742·7-s + 1.76i·13-s − 1.91i·19-s + 0.614i·31-s + 0.702i·37-s − 0.523·43-s − 0.448·49-s − 1.98·61-s + 1.99·67-s − 0.630i·73-s + 0.876i·79-s − 1.31i·91-s − 1.72i·97-s + 0.672·103-s + 0.651·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9012600592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9012600592\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.19T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 23iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 36.3iT - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 19.0iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 121T + 3.72e3T^{2} \) |
| 67 | \( 1 - 133.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 69.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 167iT - 9.40e3T^{2} \) |
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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
−8.316111824344273224234112282626, −7.18078977407144980885736755044, −6.76405562316319421309050841599, −6.16850977900883289674454642393, −4.97939654996058396601071585687, −4.44842730249761228008629461394, −3.43484726183379107926207900185, −2.59943266511814104932078073925, −1.57544673586998709744611953843, −0.23114240241234467007772449153,
0.877540448043827109386366922355, 2.12839241449394702890128758737, 3.23855689883551973946044469794, 3.67580082421201406251058839177, 4.84506435127784453957142910575, 5.80635790716562986586659237748, 6.08504286954750050120054076621, 7.18702630081911516900366872150, 7.921830714555138893440082058543, 8.374447774865867525730761387944
